0
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.