6. Let X be a random variable taking values in (0,∞) with proba- bility density function fx(u) = 5e5u u > 0. Total marks 8 Let Y = X2. Find the probability density function of Y. [8 Marks]

6. Suppose X is a continuous random variable with a strictly positive probability density function f(r); and let F(r) be the corresponding cumulative distribution function. Let U be a random variable uniform on the interval (0, 1). Show that Y = F-1(U) has the same distribution as X. Mathematical Statistics Midterm Part One Page 3 of 4

Let X be a random variable with uniform distribution on the interval [-2,2]. Let Y be defined as Y = X5. Calculate the pdf of Y.

Let X be a Gaussian random variable with zero mean and variance equal to 2. 1-Find the probability density function of the random variableY = 4X + 4 2-Find the probability P(Y>4).