Math Problem Statement

If Y has a binomial distribution with n trials and probability of success p, show that the moment-generating function for Y is m(t) = (pe^t + q)^n , where q = 1 − p

Solution

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Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Moment-Generating Function
Expected Value

Formulas

Moment-Generating Function (MGF): M_Y(t) = E[e^{tY}]
Binomial PMF: P(Y = k) = (n choose k) * p^k * (1-p)^(n-k)
Binomial Theorem: (a + b)^n = sum_{k=0}^{n} (n choose k) * a^k * b^{n-k}

Theorems

Binomial Theorem
Properties of Moment-Generating Functions

Suitable Grade Level

College/University Level