function fx,y given by Suppose that X and Y have a joint probability density Sce-2(x+y) To Find the probability P(X < 1,Y <3). fx.y (x, y) = if x > 0 and y > 0 otherwise
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- A Troublesome Snowball One winter afternoon, unbeknownst to his mom, a child bring a snowball into the house, lays it on the floor, and then goes to watch T.V. Let W=W(t) be the volume of dirty water that has soaked into the carpet t minutes after the snowball was deposited on the floor. Explain in practical terms what the limiting value of W represents, and tell what has happened physically when this limiting value is reached.The random vector (X, Y ) has the following joint probability density function:f(X,Y )(x, y) = 4xye^−(x2+y2), x > 0, y > 0,0 , otherwiseLet Z =√(X^2 + Y ^2) . Find the probability density of the random variable Z.(2) Let X1, X2, X3, X4 have the joint probability density function f(x1, x2, x3, x4) = ( 24e −(x1+x2+x3+x4) , 0 < x1, x2, x3, x4 < ∞ 0, elsewhere Let Y1 = X1, Y2 = X2 − X1, Y3 = X3 − X2, Y4 = X4 − X3. (i) Using the change of variable technique, find the joint probability density function of Y1, Y2, Y3, Y4 (ii) Find the conditional distribution of Y4 given Y1, Y2, Y3.
- Numbers X and Y between 0 and 1 are chosen randomly. The joint probability density is p(x, y) = 1 if 0 < x < 1 and 0 < y < 1, and p(x, y) = 0 otherwise. Calculate the probability P that the product XY is at least ½.A continous random variable X has the density function. f(x) = {x, 0 < x ≤ 1, {2 − x, 1 < x ≤ 2, {0, elsewhere. (a) Show that P(0 < X < 2) = 1 (b) Find P(X < 1.2)If the joint probability density of X and Y is given by:f(x, y) = 6y if 0 < x < 1, 0 < y < 1, x + y < 1 0 otherwise. a)Find marginal densities of X and Y b) Are X and Y are independent? c)Find the conditional probability of Y given that X = 1/2. d) Find covariance of r.v.’s X and Y , i.e., cov(X, Y ). Comment on their correlation based on covariancevalue.
- 1)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.Let X and Y be two continuous random variables hav-ing the joint probability density f(x, y) = 24xy for 0 < x < 1, 0 < y < 1, x + y < 10 elsewhere Find the joint probability density of Z = X + Y andW = X.Given the density function of the random variables (X,Y) as f(x,y)=3x2y+ 3xy2 , for 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 = 0 otherwise Find the marginal and conditional density functions. Also find the conditional probability P [1/2 ≤ x ≤ 3/4 | 1/2 ≤ Y ≤ 2/3]
- The joint probability density function of two random variables X and Y is f(x, y)= 4xye^-(x^2+y^2). Find P(1<x<=2, 1<Y<=2)The joint probability mass function of the two random variables (X, Y) is given by f(x,y) = { 1/5(3x-y), 1<=x<=2 , 1<=y<=3 0 , otherwise• Find the P( X ≤ Y)• Find the marginal density functions of X and Y• Are X and Y independent?• Find E(XY)The random vector (X,Y) has the following joint probability density function:f(X,Y)(x,y) ={4xye−(x^2+y^2), x >0, y >0, 0, otherwise LetZ=√(X2+Y2) Find the probability density of the random variableZ.
