Math Problem Statement

Let e_i ∼ Ber(x) , i = 1, . . . , n be identically, independently distributed. Then we have \sum_{i=1}^{n} e_i ∼ Ber(0.5 - 0.5(1-2x)^n). proof by induction.

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Bernoulli Distribution
Induction
Summation of Random Variables

Formulas

S_n = ∑_{i=1}^{n} e_i
S_n ∼ Ber(0.5 - 0.5(1-2x)^n)
Sum of independent Bernoulli random variables

Theorems

Bernoulli distribution
Principle of Mathematical Induction

Suitable Grade Level

Graduate Level (Advanced Probability Theory)