Homework SourseProblem 4 A student is walking along the real line trying to
Problem 4 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 at location x, the next move has a mean of 0 and a variance of alpha x^2. Let Xn denote the position of the student after n steps. Let N Poisson( lambda ) Find: a) E(XN |X0 = x0) b) V ar(XN|X0 = x0) Hint: Express XN as a sum of N random variables. 
Solution
sol)
Let XN denote the poisson of the student after N stepts. N~P(lambda)
XN = x1 + x2 + . . . . + xn
a) E ( Xn | X0 = x0 )
= E ( x1 + x2 + . . . . + xn | X0 = x0 )
Since X1, X2...... are independent then we can write
= E ( X1 | X0 = x0 ) + E ( X2 | X0 = x0 ) +. . . . . . . . .+ E ( Xn | X0 = x0 )
= +......
E ( Xn | X0 = x0 )=n = n(0) =0
Since mean=0 given in the statement
b)
V( Xn | X0 = x0 )
var(X1+ +XN) = var(E(X1+ +XN N)) + E(var(X1+ +XNN))
= var (0) + E (N)
= 0+2E(N)
=2
= 3
V( Xn | X0 = x0 )= 3
