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
f
Y∣X
(y∣x)=
x
1
,0≤y≤x
Find the PDF
f
Y
(y)
of
Y
. It will take the form
f
Y
(y)=
⎩
⎨
⎧
aln(b)
aln(c/y)
0
y∈[d,e]
y∈[e,f]
otherwise
Answer by finding
a,b,c,d,e,f
, where
d<e<f
Recall: If
0≤a<b
, then
∫
a
b
x
1
dx=ln(
a
b
)
. (Enter your answers as fractions, or decimals accurate to at least 4 decimal places.)
a=
Problem 3(b) 3.0 points possible (graded, results hidden) Continue from the problem above, i.e.
X
and
Y
are defined as above. Calculate
P(∣X−Y∣≤1)
(Enter an exact expression or in decimal accurate to at least 3 decimal places. To be accurate enough, use exact answers from the previous part.)
P(∣X−Y∣≤1)=
Calculate
P(min(X,Y)≤6)
(Enter an exact expression or in decimal accurate to at least 3 decimal places.)
P(min(X,Y)≤6)=
1 answer