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

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?

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.
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).