Two random variables X and Y have a joint density. f_{x,y}(x,y)=frac(10)(4) [u(x)-u(x-4)] u(y)y^3 exp [-(x+1)y^2]

Aneeka Hunt

Aneeka Hunt

Answered question

2021-09-07

4.3-16. Two random variables X and Y have a joint density
fx,y(x,y)=104[u(x)u(x4)]u(y)y3exp[(x+1)y2]
Find the marginal densities and distributions of X and Y.

Answer & Explanation

odgovoreh

odgovoreh

Skilled2021-09-08Added 107 answers

Step 1
It is given that:
fxy(x,y)=104[u(x)u(x4)]u(y)y3exp[(x+1)y2]
The marginal pdf of X is given by:
fx(X)=yxy(x,y)dy
=104[u(x)(x4)]u(y)y3exp[(x+1)y2]dy
=104[u(x)u(x4)]e(x+1)u(y)y3ey2dy
Since the explicit form of the function and its range is not given, the integral cannot be solved further.
Step 2
The marginal density of Y is given by:
fy(Y)=xfxy(x,y)dx
=104u(y)y3exp[(x+1)y2][u(x)u(x4)]dx
=104u(y)y3ey2[u(x)u(x4)]e(x+1)dx
Since the explicit form of the function and its range is not given, the integral cannot be solved further.
Step 3
Hence the marginal densities are:
fx(X)=104u(x)u(x4)e(x+1)u(y)y3ey2  and  
fy(Y)=104u(y)y3ey2[u(x)u(x4)]e(x+1)dx

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Multivariable calculus

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?