please do questions 2, 4 and 6.

PROBLEM SET 24.9 1. Let f(x, y) = k when 8 5 x = 12 and 0 S y $ 2 and 12. If the mean weight of certain (empty) containers is 5 lb zero elsewhere. Find k. Find P(X = 11, 1 s Y = 1.5) the standard deviation is 0.2 lb, and if the filling of the and P(9 = X = 13, Y S 1). containers has mean weight 100 1b and standard 2. Find P(X > 4, Y > 4) and P(X = 1, YS 1) if (X, Y) deviation 0.5 lb, what are the mean weight and the has the density f(x, y) = 2 ifx 2 0, y 2 0, x + y = 8. standard deviation of filled containers? 3. Let f(x, y) = kifx > 0, y > 0, x + y Y) when (X, Y) has the density wise. Find k. Sketch f(x, y). Find P(X + YS 1), P(Y > X'). Ax, y) = 0.25e-03(x+"> if x20,y20 4. Find the density of the marginal distribution of X in Prob. 2. and 0 otherwise. 5. Find the density of the marginal distribution of Y in 14. An electronic device consists of two components. Let Fig. 524. X and Y [years] be the times to failure of the first and second components, respectively. Assume that (X, Y) 6. If certain sheets of wrapping paper have a mean weight has the density f(x, y) = 4e-212+ ifx > 0 andy > 0 of 10 g each, with a standard deviation of 0.05 g, what and 0 otherwise. (a) Are X and Y dependent or are the mean weight and standard deviation of a pack independent? (b) Find the densities of the marginal of 10,000 sheets? distributions. (c) What is the probability that the first 7. What are the mean thickness and the standard deviation component will have a lifetime of 2 years or longer? of transformer cores each consisting of 50 layers of 15. Give an example of two different discrete distributions sheet metal and 49 insulating paper layers if the metal that have the same marginal distributions. sheets have mean thickness 0.5 mm each with a standard deviation of 0.05 mm and the paper layers 16. Prove (2). have mean 0.05 mm each with a standard deviation of 17. Let (X, Y) have the probability function 0.02 mm? f(0, 0) = /(1 1) = 8. Let X [cm] and Y [cm] be the diameters of a pin and hole, respectively. Suppose that (X, Y) has the density f(0, 1) = f(1, 0) = Are X and Y independent? f(x, y) = 625 if 0.98