Math Problem Statement

the generating function of a discrete random variable is GX (t) = 1 − √1 − t, find the probabilities for P(X = k) for k = 1, 2, . . .. Explain why P(X = 0) = 0.

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Generating Functions
Series Expansion

Formulas

GX(t) = 1 - √(1 - t)
GX(t) = Σ P(X = k) t^k for k = 0 to ∞
√(1 - t) = Σ (-1)^n * (1/2 choose n) * t^n

Theorems

Binomial Series Expansion

Suitable Grade Level

Undergraduate Level (Probability/Statistics Course)