2. Suppose a r.v. X has the density function 2 x, for 0<x<1 f(x) = 10, otherwise Observe X independently for three times, let y denote the number of an event {X<0.5) occurring in three times. (1) What is the probability of the event {X<0.5}? (2) What is the probability distribution of Y ? Write out its probability mass function
Question-2 Consider the joint uniform density function C for 22 + y2 < 4, f(x,y) 0 otherwise. What is the value of c? 0 What is P(X<0)? What is P(X <0, Y <0)? What is f( x | y=1)?
11. Let X be a continuous random variable with density function fare-102 for 10 f(1) = lo otherwise where a > 0. What is the probability of X greater than or equal to the mode of X?
4. Suppose two random variables X and Y has the following joint density function Cry, 22 Sy<1, f(x,y) = { 0, otherwise. (a) Find the constant C. (b) Find E(Y|X = 1/2). 5. Suppose X1, X2, ..., are i.i.d. random variables coming from the N(0,0%) population. (a) Determine the mean and variance for X. (b) Show that va bos (x2) – 1o60*) $ (0.2).
22. Given a continuous random variable X with probability density function f(x) = {2x, if :05451 otherwise a. Find P(0.3< X< 0.6) b. Find the mean of X C. Find the standard deviation of X.
1. Let X and Y have the joint density function given by zob to todos f(x, y) = {kxy) of 50<x< 2, 0 <y<3.) i 279VHb yodmu : 1093 otherwise a) Find the value of k that makes this a probability density function. TO B 250 b) Find the marginal distribution with respect to y. 0x11 sono c) Find E[Y] d) Find V[Y]. X10 sulay boso 50
11. The joint density function of X and Y is given by le(s+u) 0<x< o0,0<y<0 fla,y) = 01 %3D otherwise Find the density function of the random variable [Hint Use the distribution function of
(4) Suppose that the joint density function of X, Y and Z is given by )<y <<< 1 f(x, y, z) = { otherwise. (a) Find the marginal density fz(z) (b) Find the marginalized density fxy(x, y) 72 (c) Find E (2)
b. Let X be a continuous random variable with probability density function f(x) = kx2 if – 1 < x < 2 ) otherwise Find k, and then find P(|X| > 1/2).
2) Suppose that X has density function f(a)- 0, elsewhere Find P(X < .3|X .7).