Math Problem Statement

suppose X is a geometric random variable with P (X = x) = (.5)^(x+1), x ∈ {0, 1, 2, . . .}. (i) Find the generating function GX (t). (ii) If Gn(t) is the generating function for Zn where Z0 = 1 and Zn = X1 + · · · + XZn−1 , show that Gn(t) = Gn−1(GX (t)).

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Geometric Distribution
Generating Functions
Recurrence Relations

Formulas

Generating function formula: G_X(t) = Σ P(X = x) * t^x
Geometric series formula: Σ r^x = 1 / (1 - r) for |r| < 1
Recurrence relation: Gn(t) = Gn−1(GX(t))

Theorems

Law of Total Expectation
Geometric Series Summation

Suitable Grade Level

Undergraduate (Math or Probability Theory Course)