In: Statistics and Probability
(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:-
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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