Problem 4 (5 points) a. (2 points) Suppose X ~ Gam(2,0.5). Compute the probability P(4 <...
6. (5 marks) Suppose Y is a Normal random variable. If P(Y <1)=0.25 and P(Y > 0)=0.9. what are y, and o?
1. The probability distribu able X is given as follows: ition function for a certain random vari- f(0) = Tox<0 4 302 0 < < 1 > 1. (a) Find P(x > 3). (b) Find the cdf, F(x). (c) Find the mean and variance.
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
4. Suppose that X is a random variable having the following probability distri- bution function - 0 if r<1 1/2 if 1 x <3 1 if z 2 6 (a) Find the probability mass function of X. (b) Find the mathematical expectation and the variance of X (c) Find P(4 X < 6) and P(1 < X < 6). (d) Find E(3x -6X2) and Var(3X-4).
5. Suppose that the joint pdf of the random variables X and Y is given by - { ° 0 1, 0< y < 1 f (x, y) 0 elsewhere a) Find the marginal pdf of X Include the support b) Are X and Y independent? Explain c) Find P(XY < 1)
Random variables X and Y have the following joint probability density function, fx,y(x, y) = {c)[4] < 15.36, 1y| < 15.367 1.36} 0, 0.w. where cis a constant. Calculate P(Y – X| < 8.41).
Q3 Given a function which is only defined for 0<x<5 and was found by modeling. Use it to compute the probability P(0<x< 1) after first normalizing to find the valid density 1+X NOTE: set up and do all integrations by hand, show all work and NOTE: SOL P[0<x<1) 33% Q4 Let X be the chi-squared density with k-6 degrees of freedom. Find the second moment of X, SOL E(X)=
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
EXERCISE 1. Suppose Xi's are iid Negative Binomial(3,1/4) (a) Compute P(X1 < 5); (b) approximate P(21.9 X; < 1300) (c) descrive the event whose probability is computed in Part (6).