A student is walking along the real line trying to get to th

A student is walking along the real line trying to get to the origin. Each step the student makes is random;the larger the intended step, the greater the variance is of that intended step. When the student is atlocation x, the next move has a mean of 0 and a variance of . Let Xn denote the position of thestudent after n steps. Let N Poisson().

Find: a) E(XN |X0 = x0)

b) Var(XN |X0 = x0)

Hint: Express XN as a sum of N random variables.

Solution

Let us go as given in the hint,

X_N = x₁ + x₂ + . . . . + x_n

a) E ( X_n | X₀ = x₀ )

= E ( x₁ + x₂ + . . . . + xn | X₀ = x₀ )

This could be written as sum of individual expectations

= E ( X₁ | X₀ = x₀ ) + E ( X₂ | X₀ = x₀ ) +. . . . . . . . .+ E ( X_n | X₀ = x₀ )

Since the steps are independent,

The required sum will be : n

b) Variance can also be expressed as the sum of variances:

var(X₁+ +X_N) = var(E(X₁+ +X_N N)) + E(var(X₁+ +X_NN))

= var (0) + E (N)

= 0+²E(N)

=²

= ³

Hope this helps.