Assume that X and Y are jointly continuous random variables with joint probabili

sanuluy

sanuluy

Answered question

2021-11-01

Assume that X and Y are jointly continuous random variables with joint probability density function given by
f(x,y)=beg{cases}136(3xxy+4y) if 0<x<2 and 1<y<30     othrewiseend{cases}
Find the marginal density functions for X and Y .

Answer & Explanation

SabadisO

SabadisO

Skilled2021-11-02Added 108 answers

Step 1
The joint density function for X and Y is given below:
f(x,y)=beg{cases}136(3xxy+4y) if 0<x<2 and 1<y<30     othrewiseend{cases}
The marginal density function of X, g(x) is obtained as given below:
g(x)=13136(3xxy+4y)dy
=136(3xyxy22+4y22)13
=136(2x+16)
=236(x+8)
=118(x+8)
=beg{cases}118(x+8), for 0<x<20,   otherwiseend{cases}
The marginal density function of y, h(y) is obtained as given below:
h(y)=02136(3xxy+4y)dx
=136(3x22x22y+4xy)02
=636(1+y)
=16(y+1)
=beg{cases}16(y+1), for 1<y<30,   otherwiseend{cases}

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