Let Y Be a Bernoulli Random Variable with Success Probability $\hat { p }$

Essay

Let Y be a Bernoulli random variable with success probability Pr(Y = 1)= p, and let Y₁,..., Y_n be i.i.d. draws from this distribution. Let $\hat { p }$ be the fraction of successes (1s)in this sample. In large samples, the distribution of $\hat { p }$ will be approximately normal, i.e., $\hat { p }$ is approximately distributed N(p, $\frac { p ( 1 - p ) } { n }$ ). Now let X be the number of successes and n the sample size. In a sample of 10 voters (n=10), if there are six who vote for candidate A, then X = 6. Relate X, the number of success, to $\hat { p }$ , the success proportion, or fraction of successes. Next, using your knowledge of linear transformations, derive the distribution of X.