Question

Suppose that X is uniformly distributed on the interval [3,8] , and that given X=x , Y is uniformly distributed on the interval [0,x] . That is, the conditional PDF of Y given X=x is

fY|X(y|x)=1/x, 0≤y≤x.

Find the PDF fY(y) of Y . It will take the form

fY(y)=⎧⎩⎨aln(b) y∈[d,e]
aln(c/y) y∈[e,f]
0 otherwise.
Answer by finding a,b,c,d,e,f , where d<e<f .

Recall: If 0≤a<b, then ∫ba 1/xdx=ln(ba) .
a=?
b=?
c=?
d=?
e=?
f=?

Answer 1

Suppose that X is uniformly distributed on the interval [3,8] , and that given X=x , Y is uniformly distributed on the interval [0,x] . That is, the conditional PDF of Y given X=x is
fY|X(y|x)=1/x, 0≤y≤x.

Find the PDF fY(y) of Y . It will take the form

fY(y)=⎧⎩⎨aln(b) y∈[d,e]
aln(c/y) y∈[e,f]
0 otherwise.
Answer by finding a,b,c,d,e,f , where d<e<f .

Recall: If 0≤a<b, then ∫ba 1/xdx=ln(ba) .

Continue from the problem above, i.e. X and Y are defined as above.

Calculate P(|X−Y|≤1).
Calculate P(min(X,Y)≤6).

Answer 2

Any advice on this?

Calculate P(|X−Y|≤1).
Calculate P(min(X,Y)≤6).

Answer 3

a=1/5
b=8
c=8
d=0
e=3
f=8