Question

1. Let X and Y be continuous random variables with joint pdf of the form f(x, y) =

c(x+y) 0 < x < y < 2 and zero otherwise.

a.Find c so that f(x, y) is a joint pdf. Find the marginal pdf’s.
b.Find the joint CDF.
c.Find the conditional pdf of Y given x.

d.Find the conditional pdf of X given y. 

Answer 1

In the text below, S means integral and [ ] contain limits [low,high].

Joint PDF: 1=S[0,2]S[0,2](c(x+y)dxdy)⇒

1=cS[0,2]S[0,2]((x+y)dxdy)⇒

1=cS[0,2]((x^2/2+xy)[0,2])dy)⇒

1=cS[0,2](2+2y)dy)⇒

1=c(2y+y^2)[0,2]=8c, c=1/8

f(x,y)=(x+y)/8; P(X=x, Y=y)=(x+y)/8

marginal f[X](x)=(1/8)S((x+y)dy)=(1/8)(xy+y^2/2); applying the range for y: f[X](x)=(2x+2)/8=(x+1)/4

marginal f[Y](y)=(1/8)S((x+y)dx)=(1/8)(x^2/2+xy); applying the range for x: f[Y](y)=(2+2y)/8=(1+y)/4
 

Conditional densities:

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