Do only (b). Use convolution. Exercise 7.20. Let X have density fx(x) = 2x for 0...
5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y (b) (6 pts) Find the joint probability density function of V= X + Y U=X+Y and
5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6...
Let X and Y have joint probability density function fx,y(x,y) = e-(z+y) for 0 x and 0 y. Find (a) Pr(X=y (b) Prmin(X, Y) > 1/2) (c) Pr(X Y) d) the marginal probability density function of Y (e) E[XY].
2.9.10 Suppose X has density fX(x) = x3/4 for 0 < x < 2, otherwise fx(x) = 0, and Y has density fr (y)-5y4/32 for 0 < y < 2, otherwise fr (y)-0. Assume X and Y are independent, and let Z = X + Y (a) Compute the joint density fx.r(x. y) for all x, y e R (b) Compute the density fz(z) for 2.
2.8.14 Let X and Y have joint density fX,Y (x, y) = (x2 + y)/36 for −2 < x < 1 and 0 < y < 4, otherwise fX,Y (x, y) = 0. (a) Compute the conditional density fY|X (y|x) for all x, y ∈ R1 with fX (x) > 0. (b) Compute the conditional density fX|Y (x|y) for all x, y ∈ R1 with fY (y) > 0. (c) Are X and Y independent? Why or why not?
1. Let X be a continuous random variable with the probability density function fx(x) = 0 35x57, zero elsewhere. Let Y be a Uniform (3, 7) random variable. Suppose that X and Y are independent. Find the probability distribution of W = X+Y.
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.
Unif (0, 1) 5. Suppose U1 and...
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3. Let X have density fx () = 1+1 -1<<1. (a) Compute P(-2 < X <1/2). (b) Find the cumulative distribution Fy(y) and probability density function fy(y) of Y = X? (c) Find probability density function fz() of Z = X1/3 (a) Find the mean and variance of X. (e) Calculate the expected value of Z by (i) evaluating S (x)/x(x)dr for an appropriate function (). (ii) evaluating fz(z)dz, pansion of 1/3 (ii) approximation using an appropriate formula based...
. Let X and Y be the proportion of two random variables with joint probability density function f(r, y) e-*, 0, if, 0 < y < x < oo, elsewhere. a) Find P(Xc3.y-2). b) Are X and Y independent? Why? c) Find E(Y/X)
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?
Consider fx (x)=e*, 0<x and joint probability density function fx (x, y) = e) for 0<x<y. Determine the following: (a) Conditional probability distribution of Y given X =1. (b) ECY X = 1) = (c) P(Y <2 X = 1) = (d) Conditional probability distribution of X given Y = 4.