PSTAT 109 Lecture 15: Percentiles with F(x)

The joint probability density function of X and Y is given by

f(x,y)={e^-(x+y)   for X>0, y>0      and    0   elsewhere

A. Find the marginal density of X

B. Find the marginal density of Y

C. Find the Conditional density of X given Y

D. Are random variables X and Y independent? State the reason of your answer.

E. Find P(X<.5, y<.5)

F. Find P(X=.5, y<.5)

12. Let y = f(x) be defined by the equation y + 12x2 + xe4y = 13 near x = 1, y = 0. The value of f'(1) is (a) –9, (b) –7, (c) -5, (d) -10, (e) -13.

Find the least integer n such that f (x) is O(xn ) for each of these functions.
(a) f (x) = 2x2 + x3 log x
(b) f (x) = 3x5 + (log x)4
(c) f (x) = (x4 + x2 + 1)/(x4 + 1)
(d) f (x) = (x3 + 5 log x)/(x4 + 1)