that E{E(Y|X) = E) (3 marks) If the random variable X has p.d.f. - SXSTE f(x)...
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.
OUESTION 3 (20 MARKS) Consider the following joint p.d.f. oft random variable X and Y. f(x,y)= kaz’ye", 2>0, 0 <ysi 0, elsewhere (1) Determine the value of the constant k. (6 Marks) Computer E (2'y) (6 Marks) Determine E (z'ly). What can you say about the two random variables? (8 Marks)
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).
Suppose that X and Y are continuous random variables with the following joint p.d.f.: (a) Find fX|Y =y(x|y). (b) Calculate EX[X|Y = y] (c) Calculate VarX[X|Y = y] (d) Calculate E[Y] (e) Show that VarY [EX(X|Y = y)] = VarY [2/3Y ]. (f) Find VarX(X|Y = 1/2) (g) Find EX[X|Y = 0.2] (h) Without any calculation, what is P(X < Y )? Explain your answer. (i) Without any calculation, what is FX,Y (2,2)? Explain your answer. fxy(x, y)- o otherwise
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
Two random variables, X and Y, have joint probability density function f ( x , y ) = { c , x < y < x + 1 , 0 < x < 1 0 , o t h e r w i s e Find c value. What's the conditional p.d.f of Y given X = x, i.e., f Y ∣ X = x ( y ) ? Don't forget the support of Y. Find the conditional expectation E [...
3. (16 points) Suppose that X and Y have the following joint p.d.f. f(x,y) = for 0 < x < y,0 < y <, y 0 otherwise. Compute E[X2]y], the expectation of the conditional distribution of x2 given Y = y.
Give an example of a discrete or continuous random variable X (by giving the p.m.f. or p.d.f.) whose cumulative distribution function F(x) satisfies F(n)=1-1/n! Thank you very much! Exercise 3.40. Give an example of a discrete or continuous random variable X p.d.f.) whose the cumulative distribution function F(x) (by giving the p.m.f satisfies F(n)1 - i for each positive integer n or
1. Suppose that the p.d.f. of a random variable X is as follows: for 0<x<2, for 0 〈 x 〈 2. r for 0<< f(x) = 0 otherwise. Let Y - X (2 - X). First determine the c.d.f. of Y, then find its p.d.f. (Hint: when computing c.d.f., plotting the function Y- X(2 - X) which may help. )