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[cpp_is_king]

ohhh That's awesome. I've been trying to find values of t in which they are the same... But I'm having a hard time going about it.

I mean if they have the same y values, that means that intersect, right? when I set the y values equal, there are no solutions. unless I'm REALLY missing something huge here.

Find y in terms of x (or vice versa) for both P and Q.

When you do this, what are the resulting expressions that you end up with for the paths of P and Q?

 

[narcosis219]

Oops, did not realize there was a math help thread =/ I made a thread when I shouldn't have.

Anyway, my problem:

I'm taking a math course in complex analysis and all the proofy type stuff is going over my head.

I'll start by saying I'm in (electrical) engineering, so all this proper math language is really out of my comfort zone.

I've made it past analyticity and integration, but I started series last week and I feel completely lost.

For example, I'm having a hard time wrapping my head around this type of stuff:

"There is an N so that n > N ..." I believe that means that we're looking at a really big n such that this behavior can be seen. That's fine.

Epsilon is typically a very very small number, and I'm fine with that as well. So in this case, if we take a really large partial sum and subtract it from whatever it converges to (S and T in this case) we should get a really small number Epsilon. But Epsilon over 2? Where does the over 2 come from? Isn't Epsilon some arbitrarily small number anyway? What good will dividing by 2 do?

Also, I'm not entirely sure how the second line = Epsilon implies the third line.

Halp me gaf

 

[cpp_is_king]

Oops, did not realize there was a math help thread =/ I made a thread when I shouldn't have.

Anyway, my problem:

I'm taking a math course in complex analysis and all the proofy type stuff is going over my head.

I'll start by saying I'm in (electrical) engineering, so all this proper math language is really out of my comfort zone.

I've made it past analyticity and integration, but I started series last week and I feel completely lost.

For example, I'm having a hard time wrapping my head around this type of stuff:

"There is an N so that n > N ..." I believe that means that we're looking at a really big n such that this behavior can be seen. That's fine.

Epsilon is typically a very very small number, and I'm fine with that as well. So in this case, if we take a really large partial sum and subtract it from whatever it converges to (S and T in this case) we should get a really small number Epsilon. But Epsilon over 2? Where does the over 2 come from? Isn't Epsilon some arbitrarily small number anyway? What good will dividing by 2 do?

Also, I'm not entirely sure how the second line = Epsilon implies the third line.

Halp me gaf

Epsilon can be anything. In particular, whatever you decide you want epsilon to be, you could have just as easily chosen epsilon to be that number divided by 2. Since having the / 2 makes the algebra more convenient, they chose it that way up front and skipped the extra step. The reason I say it makes it more convenient is because in the last step, they end up with e/2 + e/2 = e. As you said, it doesn't matter either way since it's arbitrarily small, but this way you get the nice expression that |Sum - (S+T)| <= epsilon.

To go from line 2 to line 3, delete all the junk in the middle. What you're left with is "Sum - (S + T) <= epsilon". Since epsilon can be artibrarily close to 0 and this is still true, that means the limit of the entire left side goes to 0. So, Sum - (S+T) = 0 --> Sum = S+T

 

[narcosis219]

Epsilon can be anything. In particular, whatever you decide you want epsilon to be, you could have just as easily chosen epsilon to be that number divided by 2. Since having the / 2 makes the algebra more convenient, they chose it that way up front and skipped the extra step. The reason I say it makes it more convenient is because in the last step, they end up with e/2 + e/2 = e. As you said, it doesn't matter either way since it's arbitrarily small, but this way you get the nice expression that |Sum - (S+T)| <= epsilon.

To go from line 2 to line 3, delete all the junk in the middle. What you're left with is "Sum - (S + T) <= epsilon". Since epsilon can be artibrarily close to 0 and this is still true, that means the limit of the entire left side goes to 0. So, Sum - (S+T) = 0 --> Sum = S+T

Ah, I see. I always found it weird to see fractions of epsilon in places but it makes sense if it makes everything else cleaner.
Thanks

 

[Memento_Mori15]

Okay, I tried googling what a frequency function means in probability but getting nothing.
I have this problem:

A fair coin is tossed four times. Consider the following random variables on S, the sample space :X= the number of Heads ; Y= the length of the longest block of successive Tails (0 if NO Tails);Z= the number of the toss on which the last Tail occurred (0 if NO Tails);W=max(X; Y);V=min(X; Z). Use equally likely probability on S
(c) Find the frequency function of each random variable and plot it.

I have each variable solved. What is a frequency function?

 

[luoapp]

What is a frequency function?

    1/16, x=0,4
f=  4/16, x=1,3
    6/16, x=2

 

[Memento_Mori15]

    1/16, x=0,4
f=  4/16, x=1,3
    6/16, x=2

I didn't want the answer, just wanted to know what it meant by frequency/mass function. But you just list them out?

 

[cpp_is_king]

I didn't want the answer, just wanted to know what it meant by frequency/mass function. But you just list them out?

A function in general maps values to other values. f(0) = 1, f(1) = 2, f(3) = pi/4 is a function.

So "frequency function" isn't some kind of a standard term. It just wants a function which, for each input, tells you the corresponding "frequency" (aka probability).

So let's take the frequency function for random variable X. How often will X = 0? (1/2)^4 times. How often will X = 1? Do this for each value of X. The result is the frequency function.

 

[Memento_Mori15]

A function in general maps values to other values. f(0) = 1, f(1) = 2, f(3) = pi/4 is a function.

So "frequency function" isn't some kind of a standard term. It just wants a function which, for each input, tells you the corresponding "frequency" (aka probability).

Thank you as always.

 

[narcosis219]

More series stuff:

a) So given that sin(z) is represented by that series, can I just simply say that the power series of sin(z)/z is the same series but instead of z^(2j+1) it's just z^(2j) since I divide the whole thing by z?

b) I'm not entirely sure how to verify that the function is entire. Do I have to go back to the stuff regarding analyticity and show that the first partial derivatives are equal to each other? Or is there some other theorem that I can conveniently use?

EDIT: One of the theorems states that the series is analytic if the series is made up of analytic functions (which it is) and it converges uniformly. I'm still not too clear on uniform convergence. Does that mean that if I have a series, the next element in that series must be smaller in modulus than the last? In terms of sinx/x, I know that it looks like a decaying sinusoid that converges to 0 for -inf to inf, so going by that, it seems to have a sinusoidal behavior rather than something like an exponential decay. I have no idea what sinz/z looks like though.

c) Is it safe to just take the derivative as if it were a normal function? i.e. multiply by the power and reduce the power by 1? If I take the derivative of z^(2j) 3 or 4 times then I'll end up with z^(2j-3) and z^(2j-4) and for j = 0 and 1 it will leave me with negative powers. Is that an issue? I'm concerned since the original series only had positive powers.

I hate to ask for so much help on homework but I'm really struggling with this series stuff.

 

[cpp_is_king]

More series stuff:



a) So given that sin(z) is represented by that series, can I just simply say that the power series of sin(z)/z is the same series but instead of z^(2j+1) it's just z^(2j) since I divide the whole thing by z?

b) I'm not entirely sure how to verify that the function is entire. Do I have to go back to the stuff regarding analyticity and show that the first partial derivatives are equal to each other? Or is there some other theorem that I can conveniently use?

EDIT: One of the theorems states that the series is analytic if the series is made up of analytic functions (which it is) and it converges uniformly. I'm still not too clear on uniform convergence. Does that mean that if I have a series, the next element in that series must be smaller in modulus than the last? In terms of sinx/x, I know that it looks like a decaying sinusoid that converges to 0 for -inf to inf, so going by that, it seems to have a sinusoidal behavior rather than something like an exponential decay. I have no idea what sinz/z looks like though.

c) Is it safe to just take the derivative as if it were a normal function? i.e. multiply by the power and reduce the power by 1? If I take the derivative of z^(2j) 3 or 4 times then I'll end up with z^(2j-3) and z^(2j-4) and for j = 0 and 1 it will leave me with negative powers. Is that an issue? I'm concerned since the original series only had positive powers.

I hate to ask for so much help on homework but I'm really struggling with this series stuff.

I don't remember well enough to answer part b. For part a) your method is correct, and for part c), remember what you learned in regular calculus about trigonometric functions. Their derivatives have interesting properties.

 

[Leezard]

More series stuff:



a) So given that sin(z) is represented by that series, can I just simply say that the power series of sin(z)/z is the same series but instead of z^(2j+1) it's just z^(2j) since I divide the whole thing by z?

b) I'm not entirely sure how to verify that the function is entire. Do I have to go back to the stuff regarding analyticity and show that the first partial derivatives are equal to each other? Or is there some other theorem that I can conveniently use?

EDIT: One of the theorems states that the series is analytic if the series is made up of analytic functions (which it is) and it converges uniformly. I'm still not too clear on uniform convergence. Does that mean that if I have a series, the next element in that series must be smaller in modulus than the last? In terms of sinx/x, I know that it looks like a decaying sinusoid that converges to 0 for -inf to inf, so going by that, it seems to have a sinusoidal behavior rather than something like an exponential decay. I have no idea what sinz/z looks like though.

c) Is it safe to just take the derivative as if it were a normal function? i.e. multiply by the power and reduce the power by 1? If I take the derivative of z^(2j) 3 or 4 times then I'll end up with z^(2j-3) and z^(2j-4) and for j = 0 and 1 it will leave me with negative powers. Is that an issue? I'm concerned since the original series only had positive powers.

I hate to ask for so much help on homework but I'm really struggling with this series stuff.

For b) I think that if you have a function made up of analytic functions and you don't have any division by zero in the region, then it should be enough. Since you get that Sin(z)/z = 1 at the origin, you remove the only non-analytic part of the functions. It was a while since I did this properly, though.

 

[narcosis219]

I don't remember well enough to answer part b. For part a) your method is correct, and for part c), remember what you learned in regular calculus about trigonometric functions. Their derivatives have interesting properties.

I know that if I look at the function as sinz/z, I can just derive it and get cosine and sine terms that derive nicely, but I'm getting the idea that I'm supposed to do it in terms of the power series representation instead. In which case I have z^j which doesn't have that nice cyclical nature that trig functions do.

 

[cpp_is_king]

I know that if I look at the function as sinz/z, I can just derive it and get cosine and sine terms that derive nicely, but I'm getting the idea that I'm supposed to do it in terms of the power series representation instead. In which case I have z^j which doesn't have that nice cyclical nature that trig functions do.

Oops I didn't notice that f(z) was sin(z)/z, I thought it was just sin(z).

In any case, it's perfectly fine to compute the derivative of sin(z)/z and then deduce the power series as a result. It should work out to the same thing either way though. So to answer your quesetion: Yes you're fine just differentiating the series multiple times. Remember that each time you do this, the first term of the summation will vanish. So your final summation should start with j=3 or j=4, respectively for the 3rd and 4th derivatives. Then you can do a change of variables to transform this back to j=0

 

[Exuro]

Doing some discrete, kind of lost on this. I was given a small overview of the Euclidean Algorithm to find the greatest common divisor.

Show that the least common multiple g of n and m satisfies the formula g = nm/d where d is the greatest common divisor of n and m.

I don't know how to express the least common multiple so I have no idea what to do.

 

[Angelus Errare]

I totally read this as "The Meth Help Thread". I was about to say...you can find a thread on anything on NeoGAF.

 

[Partial Gamification]

Doing some discrete, kind of lost on this. I was given a small overview of the Euclidean Algorithm to find the greatest common divisor.

Show that the least common multiple g of n and m satisfies the formula g = nm/d where d is the greatest common divisor of n and m.

I don't know how to express the least common multiple so I have no idea what to do.

Are you looking for a proof, or just verification through a property you've perhaps covered? The least common multiple (lcm) is given, g.

m|g and n|g ('m divides g' and 'n divides g'); and, m|g and n|g are relatively prime.

What about something simple like this:
if
g = lcm(m,n)
d = gcd(m,n)
and (multiply the equation by d)
g*d = m*n
there is a property
lcm(m,n)*gcd(m,n) = m*n

There a number of ways to look at this, I could be missing something obvious.

 

[Exuro]

Are you looking for a proof, or just verification through a property you've perhaps covered? The least common multiple (lcm) is given, g.

m|g and n|g ('m divides g' and 'n divides g'); and, m|g and n|g are relatively prime.

What about something simple like this:
if
g = lcm(m,n)
d = gcd(m,n)
and (multiply the equation by d)
g*d = m*n
there is a property
lcm(m,n)*gcd(m,n) = m*n

There a number of ways to look at this, I could be missing something obvious.

I think you're on the right track. The question is kind of vague but I don't believe I have to prove it, rather verify. The notes this section is from is only over using the euclidean algorithm to find gcd. It's completely separate from our textbook so there aren't really any properties that apply to this other than the algorithm.

I have another problem dealing with strong induction that I'm struggling at.

"Suppose that x is a real number, x =/= 0 and x +1/x is an integer. Prove that for all n >= 1, x^n + 1/x^n is an integer"

First step is to use the base case which would be

P(1) = x + 1/x which is an integer so its true.

From there I'm not quite sure how the induction step works in strong induction. The textbook I have is very hard to read and the professor did not do very many examples in class.

 

[Memento_Mori15]

Sorry, but I must post another probability question. But this time, I've only brought one of them!

E, the experiment, is take a required course. The outcomes of the grade are A, B+, B, C+, C, D, F. E is repeated 5 times for each required course. Assume the grades are equally likely.
Let X count number of distinct grades you receive in the 5 courses. A, B+, B, D, and F are each given with probabilitty 1/10. C+ is given with probability 2/10, and C is given with probability of 3/10

I tried counting it without putting the probability they gave us. But when X is 2...I got stuck.

 

[cpp_is_king]

Sorry, but I must post another probability question. But this time, I've only brought one of them!


I tried counting it without putting the probability they gave us. But when X is 2...I got stuck.

I'm confused, for a number of reasons:

1) It says the grades are equally likely, then it says A, B+, B, D, and F are 1/10, C+ is 2/10, and C is 3/10. So which is it? Are they equally likely or not?

2) There's no actual question anywhere, what are you trying to find?

 

[Partial Gamification]

I think you're on the right track. The question is kind of vague but I don't believe I have to prove it, rather verify. The notes this section is from is only over using the euclidean algorithm to find gcd. It's completely separate from our textbook so there aren't really any properties that apply to this other than the algorithm.

Since you are given the gcd, and Euclid finds this, my thought was to reverse the algorithm but then you start flirting with Bezout's identity which doesn't hold for integral fields.

I have another problem dealing with strong induction that I'm struggling at.

"Suppose that x is a real number, x =/= 0 and x +1/x is an integer. Prove that for all n >= 1, x^n + 1/x^n is an integer"

First step is to use the base case which would be

P(1) = x + 1/x which is an integer so its true.

From there I'm not quite sure how the induction step works in strong induction. The textbook I have is very hard to read and the professor did not do very many examples in class.

P(n,x) = x^n + x^-n

P(1,2) = 2^1 + 2^(-1) = 5/2 not in Z. By contradiction, P(n,x) is not always in Z.
If x is in R/{0}, what if x equals; pi, e, or another irrational?

Let's change this for demonstrative purposes.
x in Q/{0}
n in N
Claim: P(n,x) is in Q.
Proof by Induction.

Base step, n=1
P(1,x) = x + x^(-1) = (x^2+1) / x is in Q, show numerator & denominator in Z/{0}

Inductive Step: fix n
Assume P(n,x) is in Q
Thus, P(n+1,x) = x^(n+1) + x^(-n-1)
= ( x^(2*n+2) + 1 ) / x^(n+1) and is in Q, show numerator & denominator in Z/{0}.

Showing the first and 'last' steps hold forces all the middle terms to follow suit. This might not be the best example to practice with.
edit: notation
Z = integers
Q = Rationals
R = Reals
Z/{0} integers, excluding zero
R/Q irrational

 

[Memento_Mori15]

I'm confused, for a number of reasons:

1) It says the grades are equally likely, then it says A, B+, B, D, and F are 1/10, C+ is 2/10, and C is 3/10. So which is it? Are they equally likely or not?

2) There's no actual question anywhere, what are you trying to find?

1)Equally likely was for the first part of the problem. It's no longer equally likely.

2) we're computing the frequency function.

Edit:

So let's take the frequency function for random variable X. How often will X = 0? (1/2)^4 times. How often will X = 1? Do this for each value of X. The result is the frequency function.

Just saw the edit of your last post. How did you get (1/2)^4?

 

[Exuro]

Since you are given the gcd, and Euclid finds this, my thought was to reverse the algorithm but then you start flirting with Bezout's identity which doesn't hold for integral fields.

P(n,x) = x^n + x^-n

P(1,2) = 2^1 + 2^(-1) = 5/2 not in Z. By contradiction, P(n,x) is not always in Z.
If x is in R/{0}, what if x equals; pi, e, or another irrational?

Let's change this for demonstrative purposes.
x in Q/{0}
n in N
Claim: P(n,x) is in Q.
Proof by Induction.

Base step, n=1
P(1,x) = x + x^(-1) = (x^2+1) / x is in Q, show numerator & denominator in Z/{0}

Inductive Step: fix n
Assume P(n,x) is in Q
Thus, P(n+1,x) = x^(n+1) + x^(-n-1)
= ( x^(2*n+2) + 1 ) / x^(n+1) and is in Q, show numerator & denominator in Z/{0}.

Showing the first and 'last' steps hold forces all the middle terms to follow suit. This might not be the best example to practice with.
edit: notation
Z = integers
Q = Rationals
R = Reals
Z/{0} integers, excluding zero
R/Q irrational

I emailed my professor about the second problem last night and she said to multiply x + 1/x with x^n + 1/x^n. Not sure how that helps. Going to read through your explanation and see if it helps.

 

[Partial Gamification]

I emailed my professor about the second problem last night and she said to multiply x + 1/x with x^n + 1/x^n. Not sure how that helps. Going to read through your explanation and see if it helps.

I'm not sure how that helps either, maybe another can add to this problem's solution -I'm at a loss.

P(1,x)*P(n,x) = x^(n+1) + x^(1-n) + x^(n-1) + x^(-n-1)

Double check the problem, I just don't think x can be in R/{0} with P(n,x) in Z.

If x = e, and a is an algebraic number, then e^a is transcendental by the Lindemann–Weierstrass theorem. The sum of the four terms could be transcendental but I'm certain its irrational.

let n = 1
x^(1+1) + x^(1-1) + x^(1-1) + x^(-1-1) = e^2 + e^0 +e^0 +e^-2
or 2 + e^2 + e^-2, clearly not an integer.

Hope I'm not leading you astray but the suggestion from your professor confuses me. Again, maybe a wiser gaffer can shine a little more light here.

 

[Leezard]

How about multiply (x + 1/x) with (x + 1/x), we get x^2 + 2 + 1/x^2, which must be an integer since its the square of an integer. Remove the 2 and the rest must also be an integer. Multiply by (x + 1/x) again and continue the argument. Make an inductive proof of this.
Edit: oh, I may be late.

 

[cpp_is_king]

1)Equally likely was for the first part of the problem. It's no longer equally likely.

2) we're computing the frequency function.

Edit:

Just saw the edit of your last post. How did you get (1/2)^4?

I'm still getting mixed up between these two apparently different questions. The (1/2)^4 was for the one about flipping coins right? In particular, this one:

A fair coin is tossed four times. Consider the following random variables on S, the sample space :X= the number of Heads ; Y= the length of the longest block of successive Tails (0 if NO Tails);Z= the number of the toss on which the last Tail occurred (0 if NO Tails);W=max(X; Y);V=min(X; Z). Use equally likely probability on S
(c) Find the frequency function of each random variable and plot it.

So the random variable X represents the number of times you flipped heads.

If X=0 that means you flipped heads 0 times. So you flipped tails 4 times. Each tails has probability 1/2. So X=0 occurs with frequency (1/2)^4.


Regarding the other question, I'm still not sure where the question is or what we're talking about. This is what you posted:

E, the experiment, is take a required course. The outcomes of the grade are A, B+, B, C+, C, D, F. E is repeated 5 times for each required course. Assume the grades are equally likely.
Let X count number of distinct grades you receive in the 5 courses. A, B+, B, D, and F are each given with probabilitty 1/10. C+ is given with probability 2/10, and C is given with probability of 3/10

I don't see any questions here though. Just something that says some grades are equally likely, then you define a variable X, then the grades aren't equally likely anymore. I think the problem is just that there's information missing here, but without a more clear write-up of the problem it's hard to answer.

 

[cpp_is_king]

I emailed my professor about the second problem last night and she said to multiply x + 1/x with x^n + 1/x^n. Not sure how that helps. Going to read through your explanation and see if it helps.

Let F(x) = x+1/x

Your teacher told you to multiply by x^n + 1/x^n, which is F(x^n)

F(x) * F(x^n) = F(x^(n+1)) + F(x^(n-1))

 

[Partial Gamification]

How about multiply (x + 1/x) with (x + 1/x), we get x^2 + 2 + 1/x^2, which must be an integer since its the square of an integer. Remove the 2 and the rest must also be an integer. Multiply by (x + 1/x) again and continue the argument. Make an inductive proof of this.
Edit: oh, I may be late.

It helps me, I was hung up on x+1/x not producing an integer for the given x, when the expression is explicitly said to be an integer.

Let F(x) = x+1/x

Your teacher told you to multiply by x^n + 1/x^n, which is F(x^n)

F(x) * F(x^n) = F(x^(n+1)) + F(x^(n-1))

This is much neater. Skipping to, essentially: integer + integer = integer

You both are very helpful, thank you.

 

[Exuro]

Let F(x) = x+1/x

Your teacher told you to multiply by x^n + 1/x^n, which is F(x^n)

F(x) * F(x^n) = F(x^(n+1)) + F(x^(n-1))

How does that show that F(x^n) is an integer though? I would need to show that F(x^n) is an integer and then that since those two multiplied give us the F(x^(n+1)) + F(x^(n-1)) that it will be an integerfor all n >= 1.

EDIT: Sec, still reading.

How about multiply (x + 1/x) with (x + 1/x), we get x^2 + 2 + 1/x^2, which must be an integer since its the square of an integer. Remove the 2 and the rest must also be an integer. Multiply by (x + 1/x) again and continue the argument. Make an inductive proof of this.
Edit: oh, I may be late.

This makes things a lot more clear. Now how do I go about making a strong induction proof from this? I'm still confused on how to set them up.

 

[Memento_Mori15]

I'm still getting mixed up between these two apparently different questions. The (1/2)^4 was for the one about flipping coins right? In particular, this one:



So the random variable X represents the number of times you flipped heads.

If X=0 that means you flipped heads 0 times. So you flipped tails 4 times. Each tails has probability 1/2. So X=0 occurs with frequency (1/2)^4.

Yes, the (1/2)^4 was referring to that question.

I'm getting confused, but I thought it would be luoapp's answer since P(x=0)=1/16.

If I were to break it down to a simpler problem say just one fair coin. For a fair coin, the probability of heads is .5, same with tails. The frequency function needs to add up to 1. P(head)+P(tails)=.5+.5=1. So I thought it would just be P(no heads)+P(1 head)+(2 heads)+P(3 heads)+P(4 heads)=1. Unless I'm still misunderstanding frequency function.

Regarding the other question, I'm still not sure where the question is or what we're talking about. This is what you posted:



I don't see any questions here though. Just something that says some grades are equally likely, then you define a variable X, then the grades aren't equally likely anymore. I think the problem is just that there's information missing here, but without a more clear write-up of the problem it's hard to answer.

Perhaps it's better to show the original problem:

The one I was doing was B.

 

[cpp_is_king]

How does that show that F(x^n) is an integer though? I would need to show that F(x^n) is an integer and then that since those two multiplied give us the F(x^(n+1)) + F(x^(n-1)) that it will be an integerfor all n >= 1.

EDIT: Sec, still reading.

Rearrange:

F(x^(n+1)) = F(x^n)*F(x) - F(x^(n-1))

By strong induction, you have integer*integer - integer.


If it helps you to see, you can do a change of variables. n = m-1

F(x^m) = F(x^(m-1)) * F(x) - F(x^(m-2))

All powers on the right are less than the power on the left, so they are covered by your induction hypothesis.

 

[Kamek]

Is there an easy way to visualize/do problems like .0016/.016 in my head?

 

[cpp_is_king]

Yes, the (1/2)^4 was referring to that question.

I'm getting confused, but I thought it would be luoapp's answer since P(x=0)=1/16.

If I were to break it down to a simpler problem say just one fair coin. For a fair coin, the probability of heads is .5, same with tails. The frequency function needs to add up to 1. P(head)+P(tails)=.5+.5=1. So I thought it would just be P(no heads)+P(1 head)+(2 heads)+P(3 heads)+P(4 heads)=1. Unless I'm still misunderstanding frequency function.

(1/2)^4 = 1/16. I don't think our answers contradict. Yes, they need to add up to 1, but I've only shown the value for X=0. If you compute the values for X=1, 2, 3, 4 and add all of them up, they will indeed add to 1.

 

[Memento_Mori15]

(1/2)^4 = 1/16. I don't think our answers contradict. Yes, they need to add up to 1, but I've only shown the value for X=0. If you compute the values for X=1, 2, 3, 4 and add all of them up, they will indeed add to 1.

...Wow, feel stupid now. >.> I don't know why I didn't just take the power. But yeah, I already found the rest and they add up to 1.

 

[cpp_is_king]

...Wow, feel stupid now. >.> I don't know why I didn't just take the power. But yeah, I already found the rest and they add up to 1.

BTW, this isn't technically necessary, but as a matter of style, I prefer to express functions in terms of a closed form expression whenever possible. Listing the exact mapping isn't wrong, but it's also not very illustrative. (i.e. it doesn't really help you understand anything about the nature of the function). It's possible to get a nice expression for f(x) as well, if you're interested in trying.

 

[Partial Gamification]

Is there an easy way to visualize/do problems like .0016/.016 in my head?

Scientific notation could be helpful.
.0016 = 1.6 x 10^-3
.016 = 1.6 x 10^-2

1.6 x 10^-3 = 1.0 x 10^-1
1.6 x 10^-2


The fractions, as a product, might help too
16/10^4 * 10^3/16 = 1/10^1

a/b = a*(1/b)

 

[Exuro]

Rearrange:

F(x^(n+1)) = F(x^n)*F(x) - F(x^(n-1))

By strong induction, you have integer*integer - integer.


If it helps you to see, you can do a change of variables. n = m-1

F(x^m) = F(x^(m-1)) * F(x) - F(x^(m-2))

All powers on the right are less than the power on the left, so they are covered by your induction hypothesis.

I think I get this, but I'm still not sure if I'm writing out the induction hypothesis correctly. I'm doing a second problem and know that it works, but I just don't know if I'm proving it correctly. Could someone give me a quick overview on how to prove by induction strong vs weak? I think that's my main issue.

 

[cpp_is_king]

I think I get this, but I'm still not sure if I'm writing out the induction hypothesis correctly. I'm doing a second problem and know that it works, but I just don't know if I'm proving it correctly. Could someone give me a quick overview on how to prove by induction strong vs weak? I think that's my main issue.

Weak induction means that given a statement is true for n, it's true for n+1.


Strong induction means that given a statement is true for 0, 1, 2, ... n, it's true for n+1.

In this case, the expression you end up with is:

F(x^n) = F(x^(n-1)) * F(x) - F(x^(n-2))

So in order to prove it's true for n, you need to know that it's true for n-1 and n-2. That's the only condition that will guarantee all 3 terms in the previous expression are integers.

This means that it's not enough to prove that it's true for n=1 and then immediately apply induction (which is what you would do for a weak induction problem), you have to manually demonstrate that it's true for n=1 AND n=2 first. If it's true for n=1 and n=2, then the above expression means it's true for n=3. And if it's true for n=2 and n=3, then it's true for n=4. Etc.


See if you can take it from here.

 

[Memento_Mori15]

BTW, this isn't technically necessary, but as a matter of style, I prefer to express functions in terms of a closed form expression whenever possible. Listing the exact mapping isn't wrong, but it's also not very illustrative. (i.e. it doesn't really help you understand anything about the nature of the function). It's possible to get a nice expression for f(x) as well, if you're interested in trying.

I would have, but after that mistake I made, I just wanted to stop with probability. lol

Quick question regarding boolean algebra. If I have (A+C')', that's essentially A'C, right?

 

[Feep]

I would have, but after that mistake I made, I just wanted to stop with probability. lol

Quick question regarding boolean algebra. If I have (A+C')', that's essentially A'C, right?

That's right! DeMorgan's.

 

[Memento_Mori15]

That's right! DeMorgan's.

Thank you. Did this during the summer and already forgetting it.

 

[FUBAR McDangles]

The wording of these problems are messing me up pretty bad. I'm trying to help my girlfriend out with her homework and I know I should know how to do these but for whatever reason I'm struggling.

Person A works twice as fast as Person B. They worked together for 10 hours then Person A left. Person B worked for 14 more hours and got the rest of the job done by himself. How long would it have taken Person A to do the entire thing?

Like, I'm reading this thinking I should know this and I should get it but I'm just completely lost. I'm tired. Her notes are horrible and don't help at all, and it's just not going well. Any help, gaf?

 

[cpp_is_king]

The wording of these problems are messing me up pretty bad. I'm trying to help my girlfriend out with her homework and I know I should know how to do these but for whatever reason I'm struggling.

Person A works twice as fast as Person B. They worked together for 10 hours then Person A left. Person B worked for 14 more hours and got the rest of the job done by himself. How long would it have taken Person A to do the entire thing?

Like, I'm reading this thinking I should know this and I should get it but I'm just completely lost. I'm tired. Her notes are horrible and don't help at all, and it's just not going well. Any help, gaf?

Let A be the amount of work person A can get done in an hour.
Let B be the amount of work person B can get done in an hour.

A = 2B

In 10 hours, the amount of work they can get done together is:

10(A + B)
= 10(2B + B)
= 30B

After 14 more hours of B working by himself, he got an additional 14B work done.

total work done = 30B + 14B = 44B

Substitute back A = 2B, and you see the total work done is 22A

Thus, A would take 22 hours to do the entire job by himself.

 

[youngwerther]

We're doing improper integrals and I'm doing my first problem and not sure if I can do it right.


it's the integral from [0,3] of 1/x dx

So of course it's undefined at 0, so I replace 0 with a and then write

lim a -> 0 from [a,3] 1/x dx

So after an easy integration you just get the limit as a approaches zero of
ln(3) - ln(a)

as a approaches zero, ln(a) becomes negative infinity.

So is the answer just infinity?

 

[DEATH™]

We're doing improper integrals and I'm doing my first problem and not sure if I can do it right.


it's the integral from [0,3] of 1/x dx

So of course it's undefined at 0, so I replace 0 with a and then write

lim a -> 0 from [a,3] 1/x dx

So after an easy integration you just get the limit as a approaches zero of
ln(3) - ln(a)

as a approaches zero, ln(a) becomes negative infinity.

So is the answer just infinity?

Yep...

 

[youngwerther]

Are you saying that I need to do something creative in order to get the correct answer or that my answer is correct and that my instructor needs to ask more creative questions?

 

[DEATH™]

Are you saying that I need to do something creative in order to get the correct answer or that my answer is correct and that my instructor needs to ask more creative questions?

Nvm... that "creative part is for another thing... dunno why it ended up there... lol

But since we are talking about creative, I tried to do 1+ the integral 1/y dy from 3 to infinity... (flipped the thing to the side) meh... ln y doesnt converge....

 

[The One Who Knocks]

EDIT: Never mind, I got it, thanks anyway though!

 

[Perfect Cha0s]

I have what I believe is a fairly simple stats problem on my hands that I can't properly wrap my head around:

Given a set of data, state the type of distribution that applies to this data, and perform a goodness-of-fit test.

Data (from a sample of 100 monitors):

Number of Bad Pixels|   0    1    2    3   4   5   6   7
   Number of Screens|   46   37   10   3   1   1   1   1

Now, I'm pretty sure this should follow a geometric distribution., with expected values of 50, 25, 12.5 etc, for 0, 1, 2, etc., respectively. However, I'm having trouble wrapping my head around how to express this mathematically, since most similar problems I've done involve coin tosses, and the "Number of Bad Pixels" in that case are number of coin tosses, and "Number of Screens" is instead number of heads (or tails), with the number of coin tosses data starting at 1, instead of 0.

The coin, of course, follows an expected value function of [(1-p)^(k-1)]*np, where p is the probability (0.5), k is the number of "successes", and n is the total number of trials. A similar setup doesn't work with the screen/pixel problem, since I can't really use k=0, and I'm not sure I can work with the presumption that the chances of a pixel being stuck/dead is 50/50.

I can proceed with the goodness of fit test simply using my own assumption, but since I can't exactly put a proper function to the situation, I don't feel 100% confident about it.

 

[Helpless Hometown Motor]

I have what I believe is a fairly simple stats problem on my hands that I can't properly wrap my head around:

Given a set of data, state the type of distribution that applies to this data, and perform a goodness-of-fit test.

Data (from a sample of 100 monitors):

Number of Bad Pixels|   0    1    2    3   4   5   6   7
   Number of Screens|   46   37   10   3   1   1   1   1

Now, I'm pretty sure this should follow a geometric distribution., with expected values of 50, 25, 12.5 etc, for 0, 1, 2, etc., respectively. However, I'm having trouble wrapping my head around how to express this mathematically, since most similar problems I've done involve coin tosses, and the "Number of Bad Pixels" in that case are number of coin tosses, and "Number of Screens" is instead number of heads (or tails), with the number of coin tosses data starting at 1, instead of 0.

The coin, of course, follows an expected value function of [(1-p)^(k-1)]*np, where p is the probability (0.5), k is the number of "successes", and n is the total number of trials. A similar setup doesn't work with the screen/pixel problem, since I can't really use k=0, and I'm not sure I can work with the presumption that the chances of a pixel being stuck/dead is 50/50.

I can proceed with the goodness of fit test simply using my own assumption, but since I can't exactly put a proper function to the situation, I don't feel 100% confident about it.

Well you could do 50*(.5)^ where n is the number of bad pixels. This is based on the formula for a geometric sequence a*r^(n-1).

However, a better way to do it is to apply a linear transformation knowing that this is an exponential distribution.