Two docnete fo Random varrable X and Y have the following joint pre fro,y)= {k (2x²y...
Two docnete fo Random varrable X and Y have the following joint pre fro,y)= {k (2x²y + 2y x² - 1 x = 0, 1, 2 Dothonorge 9 = 1,3 ① bay = Cor(xy) (1) Use Mct,, tz) to check of your answer to (1) is correct
[2.5 points] If two random variables have a joint density given by, f(x, y) = k(3x + 2y) 0 for 0 < x < 2, 0 < y < 1 elsewhere (a) Find k (b) Find the Marginal density of Y. (c) Find E(Y) (d) Find marginal density X. (e) Find the probability, P(X < 1.3). (f) Evaluate fı(x|y); (g) Evaluate fi(x|(0.75))
1. Let (X, Y) X, Y be two random variables having joint pdf f xy (xy) = 2x ,0 «x « 1,0 « y« 1 = 0, elsewhere. Find the pdf of Z = Xy?
The joint pdf fr (x)) of two random variables X and Y is given by fo (x,y)=cx2y for x +y s1. Determi use them to determine whether or not the two random variables are statistically independent. ne the constant c. Determine the marginal pdfs "Ax) and f, (y) and
7. Two random variables X and Y have joint probability density function s(x, y) = $(1 – xy), 0<x< l; 0<y<l. The marginal pdfs for X and Y are respectively S(x) = {(2-x) 0<x< 1; s()= (2-y) 0<y<l. Determine the conditional expectation E(Y|X = x) and hence determine E(Y) [7] (ii) [3] Verify your answer to part (i) by calculating the value of E(Y) directly from the marginal pdf for Y. [Total 10]
Calculate the following for the random vector (XY) with joint pdf fixy)--(3/4)(x+y) if 2x<yco, -1<x<o. 1. The marginal pdf of X and the marginal pdf of Y. Are X and Y independent random variables? 2. The expected value and variance of X and Y respectively. 3. The joint cdf in the case 2x<y<0. -1<x<0. 4. The expected value of the random variable Z defined as X^2 times Y^2. 5. The covariance between X and Y. 6. The expected value and...
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,Y) have joint density f(x,y) -2x for0 <x < 1,0sys1 and 0 elsewhere. (a) Find P(xY > z) for 0szs1. Your final answer should be a function of z. (Hint: if you pick up a particular z, say,武what is the area within the unit square of 0 x 1 and 0 y 1 such that xy > z? P1.68 shows what you need to do, i.e., a double integral. Note thatz is a constant from the perspective of both...
4.2-8. Random variables X and Y are components of a two-dimensional random vector and have a joint distribution -rebol com por odili? EIS fo xy Fx y(x, y)= { x x<0 or y<0 oito ad 05x<1 and 0s y<1 05x<1 and 1s y 15x and Osy< biller 15x and 1Sy wchongolo sistemos ( 1 (a) Sketch Fxy(x, y). (b) Find and sketch the marginal distribution functions F (x) and F,0).
Let the two-dimensional random variable (X, Y) have the joint density fx.r(x, y) = 16 - x - y)I(0, 2)(x)/(2,4,(y). (a) Find &[Y| X = x]. (6) Find &[Y|X = x]. © Find var (Y|X=x]. (d) Show that &[Y] = { [E[Y|X]]. (e) Find &[XY|X=x]. Tinomial distribution (multinomial with k + 1 =3) of two random variables The trinomial distribution (mu X and Y is given by fx.x(x, y) = x!y!(n - x - y)!' for x, y=0, 1, ...,...