## Transcribed Text

1. (5pts) Suppose X and Y are random variables with joint probability density function
2vxe x+y 3
f F(x,y) =
,
X > 0,y > 0
3VII
0,
otherwise.
Are X and Y independent? Explain.
2. Suppose X and Y have the joint probability function
f(x,y)=(cl2x+ = y), 0,
+
0 < X < 1,0 < y < 2
otherwise.
a.
(5pts) Find the value of C.
b. (8pts) Find the marginal distribution of Y.
c. (10pts) Find the joint distribution function F(x,y).
3. Suppose X and Y have joint probability density function
3y
f F(x,y) =
,
< y < X < 2, ,
4
0,
otherwise.
a.
(10pts) Find P(Y < 1).
b.
(10pts) Find E(Y|X = x).
c.
(10pts) Find P(Y < 0.5|X=1).
4. Assume that X ~ Exp(1) and Y ~ Exp( (1) are two independent variables. Let S = Y and T = X + XY.
Find
a. (5pts) E (E(T|S))
b. (10pts) V(T)
5. Let Y1, Y2, , Yn be a random sample from a distribution with the probability density function
y > 1
=
otherwise,
where 0 > 0.
a. (7pts) Find the method of moments estimator of 0.
b. (10pts) Find the maximum likelihood estimator for 0.
6. (10pts) Let Y1, , Y2, ,Yn , be a random sample from the uniform distribution on the interval (0, 10 - 20)
where 10 < 0 < 5. Let 0 = 10 - 2Y. Show that 0 is consistent for 0.
10

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