B.4 Let the binary random variable X have pdf f (x) = p'( i - p)'' for x = 0. 1.

(a) Find the mean and variance of X.

(b) Let X1 ..... X, be independent discrete (0.1) random variables each with probability density function f(x). The random variable B = X: + X2 + +

X, has a binomial distribution with parameters n and p. The values of B are b = 0 ...... and represent the number of "successes" (i.c ., X¡ = 1) in "

independent trials of an experiment, cach with probability p of success. Calculate the mean and variance of B using the rules of expected values and variances.

[Hint: What is the expected value of a sum? The variance?]

(e) Let X1 .... , X„ be independent discrete (0. 1) random variables with probability density functions (x). The random variable B = X1 + X2 + . + X, has a binomial distribution. The random variable Y = B/n is the proportion of successes in n trials of an experiment. Find the mean and variance of Yusing the rules of expected values and variances.