Probability: having trouble with explanations

[terp.asessed]

Homework Statement

I GOT all values...but I have trouble explaining...if someone could suggest how, thank you! Please check the red fonts which shows where I got stuck!

Suppose we call x the position of the boy and each step he takes (east or west ONLY) is of 1m. So, for an "n" step random walk he takes, the n+1 possible positions that the boy can end up at are given by:

x = 2k +n, where k = 0, 1 ...n

and the likelihood of ending up at anyone of these is determined by the probability: p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and k = 0, 1 ... n.

(a) Explain why the formula p(k) should be true.
(b) Compute <x> after an n-step walk. Why should <x> have this value?
(c) compute [<x^2>]^0.5 after n step walk. Suppose all n steps were in the same direction, what would be [<x^2>]^0.5?

Homework Equations

x = 2k +n, where k = 0, 1 ...n
p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and C is a normalization constant and k = 0, 1 ... n.

The Attempt at a Solution

([/B]a) B/c Sigma (k=0 to n) p(k) = 1 for normalization, if n =1, supposing x = -1, k would be equal to 1...giving a p(1) = 1/2 (1/1) = 1/2. This is the explanation I put...but, I don't understand in the case when n=1, x = 1, in which k = 0, giving p(0) = 1/2 (1/0) = ?. Could someone hint why the formula is true?

(b) I obtained the value by:
<k> = Sigma (k=0 to n) k*p(k)
= Sigma k (n k) 1/(2^n)
= 1/(2^n) Sigma k (n k) = 1/(2^n) n * 2^(n-1) = n/2

therefore...<x> = 2<k> - n
= 2(n/2) - n = 0...Why did I get the value of zero, if uncertainty is possible? Is it b/c there is an equal chance of being east or west of the initial position the boy was in the first place?

(c) Similarly, I derived <k^2> which I got n(n+1)/4. So:
<x^2> = <(2k+n)^2> = <4k^2 - 4kn + n^2> = 4<k^2> - 4n<k> + n^2 into which I input the results I had from the above...and got <x^2> = n
So...[<x^2>]^0.5 = n^0.5. However, what should I do if all n steps were in the same direction? Because that is once in all possible probabilities, so, should it be 1/n^0.5? I am really stuck here!

 

[Ray Vickson]

Homework Statement

I GOT all values...but I have trouble explaining...if someone could suggest how, thank you! Please check the red fonts which shows where I got stuck!

Suppose we call x the position of the boy and each step he takes (east or west ONLY) is of 1m. So, for an "n" step random walk he takes, the n+1 possible positions that the boy can end up at are given by:

x = 2k +n, where k = 0, 1 ...n

and the likelihood of ending up at anyone of these is determined by the probability: p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and k = 0, 1 ... n.

(a) Explain why the formula p(k) should be true.
(b) Compute <x> after an n-step walk. Why should <x> have this value?
(c) compute [<x^2>]^0.5 after n step walk. Suppose all n steps were in the same direction, what would be [<x^2>]^0.5?

Homework Equations

x = 2k +n, where k = 0, 1 ...n
p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and C is a normalization constant and k = 0, 1 ... n.

The Attempt at a Solution

([/B]a) B/c Sigma (k=0 to n) p(k) = 1 for normalization, if n =1, supposing x = -1, k would be equal to 1...giving a p(1) = 1/2 (1/1) = 1/2. This is the explanation I put...but, I don't understand in the case when n=1, x = 1, in which k = 0, giving p(0) = 1/2 (1/0) = ?. Could someone hint why the formula is true?


(b) I obtained the value by:
<k> = Sigma (k=0 to n) k*p(k)
= Sigma k (n k) 1/(2^n)
= 1/(2^n) Sigma k (n k) = 1/(2^n) n * 2^(n-1) = n/2

therefore...<x> = 2<k> - n
= 2(n/2) - n = 0...Why did I get the value of zero, if uncertainty is possible? Is it b/c there is an equal chance of being east or west of the initial position the boy was in the first place?

(c) Similarly, I derived <k^2> which I got n(n+1)/4. So:
<x^2> = <(2k+n)^2> = <4k^2 - 4kn + n^2> = 4<k^2> - 4n<k> + n^2 into which I input the results I had from the above...and got <x^2> = n
So...[<x^2>]^0.5 = n^0.5. However, what should I do if all n steps were in the same direction? Because that is once in all possible probabilities, so, should it be 1/n^0.5? I am really stuck here!

The position should be $x = 2k-n$, not $2k+n$. Think about it: if he takes $k$ steps to the right and $n-k$ steps to the left, where does he end up?

You say that in (a) you do not understand the formula for n = 1. Which formula do you mean? Do you want to know why (1 0) = 1? Well, that is just the number of ways of getting 0 heads when you toss a coin one time: you get either H (once) or T (once). In other words, there is just one way to get 1 head and 1 way to get 0 heads. In general, (n 0) = 1 for any integer n because when you toss a coin n times there is just one way of getting 0 heads---namely, by getting all tails.

BTW: the standard notation for (a b) is either _aC_b or C(a,b) or $\binom{a}{b}$. You get the first one by using the "$x_2$" button on the panel at the top of the input page; you get the second one by straight typing; you get the third one using LaTeX.

In (b) you say you do not understand why $\langle X \rangle = 0$. Well, what is the MEANING of $\langle X \rangle$? It is the average value of X in a large number of identical trials of the situation under study. A mean of 0 just means that going left or right is equally likely. For example, when n = 1 you end up either at x = +1 or x = -1. If you do the experiment 2 million times, then in about 1 million of the experiments you will be at x = 1 and in about 1 million experiments you will be at x = -1. On average, then, your final location will be at zero.

In (c): the value of $\langle X^2 \rangle$ looks at the average of all the possible $x^2$ values. Only one of the very many outcomes has all steps to the right (or all to the left). The contribution of those two extreme outcomes to $\langle X^2 \rangle$ would be $2 C(n,0) n^2 / 2^n = 2 n^2/2^n$ and so is small compared with the total due to other outcomes.

 

[terp.asessed]

Hello, thank you very much!
I get (b) and (c)! Also thank you for pointing out how to get the standard notation...I get how to do first two, but, how do I use "Latex"?

Aside, regarding (a), by formula, I mean I have trouble explaining why p(k) = 1/(2ⁿ) (n k) where (n k) = n! / [(n-k)!(k!)] is true--should I verify the function with examples? Or, state that it is b/c of normalization? This is where I am confused.

 

[haruspex]

Hello, thank you very much!
I get (b) and (c)! Also thank you for pointing out how to get the standard notation...I get how to do first two, but, how do I use "Latex"?

Aside, regarding (a), by formula, I mean I have trouble explaining why p(k) = 1/(2ⁿ) (n k) where (n k) = n! / [(n-k)!(k!)] is true--should I verify the function with examples? Or, state that it is b/c of normalization? This is where I am confused.

You may also be under the misapprehension that 0! = 0. By convention, 0!=1. This makes sense because it means that the rule n!*(n+1) = (n+1)! applies even when n=0.
For p(k), break the problem into two parts. First, how many different sequences of n steps are there? Are they equally likely? What is the probability of each sequence? Next, how many sequences consist of k steps one way and n-k the other?

 

[terp.asessed]

You may also be under the misapprehension that 0! = 0. By convention, 0!=1. This makes sense because it means that the rule n!*(n+1) = (n+1)! applies even when n=0.

So, 0! = 1. Still, 1! = 1, right?
Also, about:

For p(k), break the problem into two parts. First, how many different sequences of n steps are there? Are they equally likely? What is the probability of each sequence? Next, how many sequences consist of k steps one way and n-k the other?

Since the different sequences of n steps there are are 2ⁿ, where x + n, x -n both have probability value of 1/2ⁿ, whilst the other possibilities are somewhere between two extreme distances, making up of the probability value of (2ⁿ -1)/2ⁿ.
As for k = (x + n)/2 after the boy's initial position has moved n steps...the max value k can have is when n steps were in the west direction, as in x - n = 2k -n, in this case, x = 2k...then...because k is dependent on n values, if k = n, then x = 2(n) - n = n...so, how do I get from here? Or, am I getting lost?

 

[Ray Vickson]

So, 0! = 1. Still, 1! = 1, right?
Also, about:
Since the different sequences of n steps there are are 2ⁿ, where x + n, x -n both have probability value of 1/2ⁿ, whilst the other possibilities are somewhere between two extreme distances, making up of the probability value of (2ⁿ -1)/2ⁿ.
As for k = (x + n)/2 after the boy's initial position has moved n steps...the max value k can have is when n steps were in the west direction, as in x - n = 2k -n, in this case, x = 2k...then...because k is dependent on n values, if k = n, then x = 2(n) - n = n...so, how do I get from here? Or, am I getting lost?

I cannot figure out what you are asking, but before you wanted to know why $P(k) = \binom{n}{k}/2^n$. Is that still what is bothering you? Well, look at some small-scale examples, where you can enumerate all the possibilities. Let the successive outcomes (steps) be labelled as R or L, depending on their direction.

For n = 3 the possible outcomes are RRR, RRL, RLR, RLL, LLL, LLR, LRL, LRR. There are 8 different outcomes, each having probability 1/8 = 1/2^3. We have P(3R,0L) = P(RRR) = 1/8, P(2R,1L) = P(RRL)+P(RLR) + P(LRR) = 3 * 1/8, P(1R,2L) = P(RLL) + P(LRL) + P(LLR) = 3 * 1/8, and P(0R,3L) = P(LLL) = 1/8. So, the coefficients C(3,k) just count the number of different ways of having k Rs and (3-k) Ls.

In general, for k Rs and n-k Ls, we have many different "strings", such as RR...RLL...L, RRLR...LLL LR, etc. It is not too difficult to show that the total number of distinct such strings is
$$\text{no. of strings } \; = \binom{n}{k} = \frac{n!}{k! (n-k)!} = \frac{n(n-1) \cdots (n-k+1)}{k!}$$
In general, if $p$ is the probability of a step to the right and $q = 1-p$ is the probability of a step to the left, the probability of $k$ stepes to the right in $n$ steps altogether is
$$P(k) = \binom{n}{k} p^k q^{n-k}$$
In the special case $p = q = 1/2$ one gets $P(k) = \binom{n}{k}/2^n$.

 

[terp.asessed]

Thank you very much.

One last clarification...

In (c): the value of $\langle X^2 \rangle$ looks at the average of all the possible $x^2$ values. Only one of the very many outcomes has all steps to the right (or all to the left). The contribution of those two extreme outcomes to $\langle X^2 \rangle$ would be $2 C(n,0) n^2 / 2^n = 2 n^2/2^n$ and so is small compared with the total due to other outcomes.

I think I made a mistake somewhere, for I got <x²> = n, not n²...what I did was:

<k²> = sigma (k=0 to n) k² p(k) = (1/2ⁿ) sigma k² (n,k) = 1/(2ⁿ) n(n+1)2^n-2 = n(n+1)/4
<k> as done previously = n/2
<x²> = <(2k-n)²> = 4<k²> - 4<k>n + n² = n2 + n - 2n² + n² = n

...so, I got C(n,0) n = 2 (1/2ⁿ n!/n!) = 2n/2ⁿ, NOT 2n²/2ⁿ...

Could you please point out where I did wrong? If so, merci beaucoup!

 

[Ray Vickson]

$\langle X^2 \rangle = n$ is correct.

I cannot figure out what your "C(n,0)n" business is all about, or why you should care. The fact is that C(n,0) = 1 so C(n.0)n = n. End of story.

 

[terp.asessed]

Got it!