# Question: If X1 X2 Xn are independent

If X1, X2, . . . , Xn are independent random variables having identical Bernoulli distributions with the parameter θ, then is the proportion of successes in n trials, which we denote by Θ. Verify that

(a) E(Θ) = θ;

(b) var(Θ) = θ(1 – θ)/n .

(a) E(Θ) = θ;

(b) var(Θ) = θ(1 – θ)/n .

## Relevant Questions

Verify that if T has a t distribution with v degrees of freedom, then X = T2 has an F distribution with v1 = 1 and v2 = v degrees of freedom. Verify the results of Example 8.4, that is, the sampling distributions of Y1, Yn, and X~ shown there for random samples from an exponential population. Find the sampling distribution of Y1 for random samples of size n = 2 taken (a) Without replacement from the finite population that consists of the first five positive integers; (b) With replacement from the same ...Use the result of Exercise 8.58 to show that, for the random variable P defined there, What can we conclude from this about the distribution of P when n is large? Rework part (b) of Exercise 8.66, assuming that the population is not infinite but finite and of size N = 400.Post your question