Math Problem Statement

P (N = n) = (.5)^n for n = 1, 2, . . . and S = X1 + · · · + XN , with the (Xi) independent copies of X and P (X = 0) = P (X = 1) = 1 2 . Find P (S = 0) and P (S = 1). Find the generating function GS (t). Find GS (0) and explain why it is the same as P (S = 0)

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Random Variables
Generating Functions
Binomial Distribution

Formulas

P(N = n) = (0.5)^n
P(S = k) = Σ P(S = k | N = n) P(N = n)
Generating Function: G_S(t) = E[t^S] = Σ P(S = k) t^k

Theorems

Law of Total Probability
Binomial Distribution
Generating Function Theorem

Suitable Grade Level

Graduate Level / Advanced Undergraduate