Finding mean and variance of random walk with variance proportional to position squared

[Milky]

An individual traveling on the real line is trying to reach the origin. However, the larger the desired step, the greater is the variance in the result of that step. Specifically, whenever the person is at location x, he next moves to a location having mean 0 and variance $$\beta x^2$$. Let $$X_n$$ denote the position of the individual after having taken n steps. Supposing that $$X_0 = x_0$$, find
a. $$E[X_n]$$
b. $$Var(X_n)$$.

I am not sure how to even start this problem, and would really appreciate any suggestions!

 

[EnumaElish]

Did you mean:

"whenever the person is at location x_n, he next moves to a location x_n+1 with error having mean 0"?

 

[Milky]

No, I copied exactly what the book said. I'm so confused!

 

[EnumaElish]

Okay. Part a is easy. The problem gives away the answer. (The answer is stated unambiguously as part of the problem statement.)

For part b, what does the definition of variance say? (E.g. Var[x] = b x^2.)

 

[Milky]

Okay, so for part a, E[Xn] = 0 because with every step the mean is 0...right?

For part b.. I'm still not sure.
I know Var(X) = bx^2, but I don't know how to get from X to Xn. I tried using the definition of variance: E[X^2] - (E[X])^2 but it didn't get me very far

 

[EnumaElish]

I know Var(X) = bx^2, but I don't know how to get from X to Xn.

The question is, if you know Xn, do you need to know X0, ..., X(n-1) to know Var(Xn)?

 

[Milky]

The variance is the same for each step that he takes, so why wouldn't the variance just still be bx^2?

 

[EnumaElish]

No, the variance changes with each step, unless you happen to stay where you are (which is highly improbable). Look at the definition of the variance. It maps x to bx^2. Suppose Xn = t. Where does it map t?

 

[Milky]

bXn^2 ?

 

[EnumaElish]

Yes.