OUESTION 3 (20 MARKS) Consider the following joint p.d.f. oft random variable X and Y. f(x,y)=...
that E{E(Y|X) = E) (3 marks) If the random variable X has p.d.f. - SXSTE f(x) = {20 'o, otherwise, y = ex Gly)= Prob (Ys y) = Probe Prob(ancex) sluca inly) x < lncy F(x) dx = e cumulative distribution function technique to determine the p.d.f. of Y=e (4 marks CJE marks) avoy Given that the continuous random variable X and Y have joint p.d. f. f(x,y). She
qkx Q.3 (20 pts) The joint p.d.f. of 2 random variables x and y is given by f xy(x,y) = 05x,0 sy.(2x + y) = 2 to otherwise Where k is a constant. 1) (5 pts) Find the value of k. 2) (5 pts) Are x and y independent? Explain. 3) (10 pts) Define z = 2x-y. Find the p.d.f. of z.
QUESTION 12 Let the random variable X and Y have the joint p.d.f. f(x,y) =(zy for 0< <2, 0 < y <2, and z<y otherwise Find P(0KY <1) 16 QUESTION 13 R eter to question 12. Find P(o < x <3I Y-1).
(50 points) Suppose that the joint p.d.f. of X and Y is as follows: for x 2 0, y 2 0, and x + y <1 elsewhere 2. 24xy f(x)0 a) Determine the value of P(X < Y). b) Determine the marginal p.d.f.'s for Xand Y c) Find P(X> 0.5) d) Determine the conditional p.d.f. of X|Y = 0.5 e) Find P(X> 0.5|Y 0.5) f) Find P(X> 0.5|Y> 0.5) g) Find Cov (X, Y)
QUESTION 9 Let the random variable X and Y have the joint p.d.f. f(x,y) for the (x,y) pairs as shown in the following table (for x = 0,1,2 and y = 0.1). y/X 0 1 2 0 1 14 6 | 18 18 1133 18 18 Find the covariance oxy O-57/324 O-58/324 57/324 58/324
The random variable X has the following p.d.f. tx -9 - <I< an f(3) = 21 0, otherwise Use the moment generating function technique to determine the p.d.f. of Y=X? Hence or otherwise state the mean and variance of the random variable Y. (6 Marks)
Let X and Y be random variables for which the joint p.d.f. is as follows: f (x, y) = 2(x + y) for 0 ≤ x ≤ y ≤ 1, 0 otherwise.Find the cumulative distribution function (c.d.f.) of X and Y.Find p.d.f. of Z=X+Y.
Question 3 (20 marks) Two random variables X and Y have the following joint probability density function. $(x,y)={{(6xy? +3y2 + 2x+) 0<x<2, 1<y<2, 0, otherwise Determine the value of the constant k. (5 Determine the marginal density functions and hence check if X and Y are inde Work out the difference between E{var(Y|X} and V ar{ECY|x)} (10 ho Question 4 (20 marks) ay -Express x in terms of Work our screen of /ke X VY J=dx
Let X and Y be random variables with joint density function F(x,y) O<ysi< otherwise The marginal density of Y is fr() = 3 (1 - ), for 0 < y<1. True False
Find the conditional p.d.f.’s f(y|x) and f(z|x, y). 4. Suppose that random variables (X, Y, Z) have the joint p.d.f. f(x,y,z)-' 0, otherwise . ind the conditional p.d.f.'s f(yx) and f (z x,y