Math Problem Statement
Prove, using induction, the PMF of a binomial using convolutions on the iid sequence of r.v.'s X1,...,Xn ∼ Bernoulli(p). You will need to use Pascal's Triangle combinatorial identity
Solution
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Binomial Distribution
Bernoulli Distribution
Induction
Convolutions
Pascal's Triangle
Formulas
PMF of Binomial Distribution: P(S_n = k) = (n choose k) * p^k * (1 - p)^(n - k)
Pascal's Identity: (n choose k) + (n choose k-1) = (n+1 choose k)
Convolution Formula: P(S_{n+1} = k) = P(S_n = k) * P(X_{n+1} = 0) + P(S_n = k-1) * P(X_{n+1} = 1)
Theorems
Pascal's Triangle Combinatorial Identity
Mathematical Induction
Suitable Grade Level
Advanced Undergraduate / Graduate Level