G). If the probability density of X is given by 01 Λ 1 if -1<0, f(x)-1-x,...
. Let X and Y be the proportion of two random variables with joint probability density function f(r, y) e-*, 0, if, 0 < y < x < oo, elsewhere. a) Find P(Xc3.y-2). b) Are X and Y independent? Why? c) Find E(Y/X)
The probability density function of X is given by
0 elsewhere
Find the probability density function of Y = X3
f(r)-(62(1-x)for0 < x < 1
5. Given the probability density f(x)= for -0<x<00, find k. 1+ 2 Jor -
Suppose that the error temperature of a Lab has a probability density f(x)= 1- |x|) -1<x<1 _ 0 elsewhere Using R to compute the probability that the error temperature is between 0 and 0.5.
. Let X and Y be the proportion of two random variables with joint probability density function f(x, y)o, elsewhere. (a) Find P(X < 3|Y= 2). (b) Are X and Y independent? Why? (c) Find E(Y/X)
If X and Y have a joint density given by f(x, y)- 2, for 0 < y < x < 1 0, elsewhere (a) If V - -InX, what is the density of V? (b) If V -InX and W X + Y, what is the joint density of V and W? Sketch the region for which the joint density is nonzero
(8pts) 1. The joint probability density of X and Y is given by + 0<x<1 and 0 <y< 2 otherwise a) Verify that this is a joint probability density function. b) Find P(x >Y). o) Find Pſy > for< d) Find Cov(X,Y). e) Find the correlation coefficient of X and Y (Pxy).
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
If the probability density function of X is given by n2 for 1<x< 2 fx ) = 10 elsewhere (a) Find, E[X], E[X2], and E[X3] (b) Use your answer to part (a) to find E[X3 + 3X2 - 2x + 5)
The density function of X is given by + br if 0 r < 1 f(x) = 0 1 otherwise If E(X) = 3, find a and b. (Hint: Both values are integer.) a = b =