Question

(a) Let X and Y have the joint pdf ???(?, ?)=1, 0≤x≤3/2, 0≤y≤1, zero elsewhere. Find: 1 The pdf of Z=X+Y 2 The pdf of Z=X.Y

Answer 1

Answer:-

Given That:-

Let X and Y have the joint pdf     zero elsewhere. Find

The pdf of Z = X + Y

The pdf of Z = X*Y

Given,

If X and Y have joint pdf

  

The pdf of Z = X + Y

Use the transformation,

Z = X + Y, W = Y

then,

Support of X, Y

Support of Z, W

Then,

Y = W

X = Z - Y = Z - W

then to find Jacobian,

J = 1

Hence,

The Jacobin is 1

By the formula we know that joint pdf of Z = X + Y, W = Y is

Then,

origial pdf of z is given by

  

pdf of z = x + y is given by

The pdf of Z = X*Y

To find:-

pdf of z = xy

Use the transformation

z = xy, w = y

then,

x = z/w, y = w

Again,

0 < x < 3/2, 0 < y < 1

0 < z/w < 3/2, 0 < w < 1

0 < z < 3w/2, 0 < w < 1

0 < z < 3w/2 < 3/2

then, Jacobian is

= 1/w

Then,By the formula,

Joint pdf of z and w is

Then, the marginal pdf of z is,

Hence, Marginal pdf of z is

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