Example 16-22. If F(n1, n2) represents an F-variate with n¡ and n2 d.f., prove that F(n2, n1) is distributed as 1/F(n, n2) variate. Deduce that P[F(n1, n2) 2 c] = P| F(n2, n1) < %3D

Let X be a random variable with pdf: f(x)= k(10 - 2x) for x being [0,5]  a.) Find k b.) Find E(X) c.) Find Var(X) d.) Find E(3 *squareroot(X))

8. (Same as #4) At a particular gas station, gasoline is stocked in a bulk tank each week. Let random variable X denote the proportion of the tank's capacity that is stocked in a given week, and let y denote the proportion of the tank's capacity that is sold in the same week. The joint pdf of X and Y is given by f(x,y) (3x if 0 ≤ y < x < 1 = 0 otherwise a) Find the marginal probability distributions of X and Y. b) Show that the two random variables are not independent. Using part of your answer to a) is a much shorter solution. Once you've done this with probability, using common sense is VERY straightforward.

5

If X1, X2 N(0, 1), then show the joint distribution of Y1 = aX1 + bX2+c Y2 = dX1 + eX2+f is bivariate normal.

A random variable X is uniformly distributed on the interval (- 5, 15). Another random variable Y = e¯x/5 is formed. Find E [Y] and fy (y).

Let X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1.    Find the marginal fx(x) and fy(y). Show if f(x,y) dependent.  Compute mean(x), mean(y), variance (x) and variance(y)

Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf f(x, y) =    K(x2 + y2)      19 ≤ x ≤ 29, 19 ≤ y ≤ 29 0   otherwise (a) Determine the conditional pdf of Y given that X = x. fY|X(y|x) =  for 19 ≤ y ≤ 29 Determine the conditional pdf of X given that Y = y. fX|Y(x|y) =   for 19 ≤ x ≤ 29 (b) If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? (It is known that  K =  3 350,600 .  Round your answer to three decimal places.) Compare this to P(Y ≥ 25). (Round your answer to three decimal places.)P(Y ≥ 25) =  (c) If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire? (It is known that  K =  3 350,600…

A random variable x follows continuous uniform distribution on the interval [18,38]. Find the variance V ar (X). Answer should be with 2 decimal places.

Let X1 and X, be independently and uniformly distributed over the interval (0, a). Find the p.d.f. of (a) U = X1 + X2. (b) W = X1 – X2.