1. Find the mean and variance for the distribution of random variable X whose density function is f(x). 1) = { ² x ²0 - ² 0 f(x, y) 16 Mean: x>0 elsewhere Variance:

a) Find the best point estimate of the population proportion p. b) Identify the value of the margin of error E. ​c) Construct the confidence interval.

12. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))

1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.