Suppose that a statistical model is given by the family of Bernoulli($\theta$) distributions where $θ \in \Omega = [0, 1]$. Calculate $\psi(\theta)$

Question

Suppose that a statistical model is given by the family of Bernoulli($\theta$) distributions where $θ \in \Omega = [0, 1]$. If our interest is in making inferences about the probability that two independent observations from this model are the same, then determine $ψ(θ)$.

SOLUTION: We have that $\psi(\theta) = (1-\theta)^2 + \theta^2$. How?

Answer 1

Suppose $X_1, X_2\stackrel{\text{i.i.d}}{\sim}\text{Bernoulli}(\theta)$ i.e. $P(X_1=1)=\theta$ and $P(X_1=0)=1-\theta$. Then $$ P(X_1=X_2)=P(X_1=1, X_2=1)+P(X_1=0, X_2=0)=P(X_1=1)^2+P(X_1=0)^2 $$ by independence and Identically distributed assumption. The result follows.

Answer 2

This is because we get two results the same iff both results are $0$ or both are $1$. The probability both are $0$ is $\color{blue}{(1-\theta)^2}$ and the probability that both are $1$ is $\color{blue}{\theta^2}$ (because of independence). The probability of getting same results is thus the sum of these (since "both results are $0$" and "both results are $1$" are disjoint events).