A random sample of size 8 X 1 1, X 2 0, X 3 1, X 4 1, X 5 0, X 6 1, X 7 1, and X 8 0 is taken from the probability function p X (k ) k (1 ) 1k , k 0, 1 0 1 Find the maximum likelihood estimate for ...

A random sample of size 8 X 1 = 1, X 2 = 0, X 3

Question:

A random sample of size 8 – X₁ = 1, X₂ = 0, X₃ = 1, X₄ = 1, X₅ = 0, X₆ = 1, X₇ = 1, and X₈ = 0 – is taken from the probability function p_X(k; θ) = θ^k(1 − θ)^1−k, k = 0, 1; 0 < θ < 1. Find the maximum likelihood estimate for θ.