Finding Probability Generating Function for H0

[hsong9]

Homework Statement

Let Y1,Y2,... be independent identically distributed random variables with
P(Y1=1) = P(Y1=-1) = 1/2 and set X0 = 1, Xn=X0 + Y1 +...+ Yn for n >= 1. Define
H₀ = inf { n >= 0 : Xn = 0}.
Find the probability generating function Φ(s) = E(s^H0).

Homework Equations

The Attempt at a Solution

Actually, this is grad. level course, but
I do not have much some background about probability.
I don't know whether I can get the values of X2, X3,... Xn by using Y1 =1 = Y1 = -1...
Is it possible?

 

[mighty2000]

Hello, thank you for your question. Yes, it is possible to find the values of X2, X3,... Xn by using the given information about Y1. Since Y1 is an independent identically distributed random variable, it means that each Yn has the same distribution with the same probabilities. Therefore, you can use the given probabilities of Y1 to calculate the probabilities of Y2, Y3,... Yn and then use those probabilities to calculate the values of X2, X3,... Xn. Once you have the values of X2, X3,... Xn, you can use them to calculate the probability generating function Φ(s) = E(sH0). I hope this helps.