Statistics 426: Final Exam
Moulinath Banerjee
April 28, 2004
Announcement:
The final exam carries 80 points. Half of what you score contributes to your
grade in the course.
(1) . (a) Let
Z
1
, Z
2
, Z
3
be i.i.d. N(0,1) random variables. Let
R
=
q
Z
2
1
+
Z
2
2
+
Z
2
3
. Find the
density function of
R
. (Hint: Can you write down the density function of
R
2
?)
(b) Suppose that
X
∼
Γ(
α, λ
) and
Y
∼
Γ(
β, λ
) where
α, β, λ >
0. Let
U
=
X
+
Y
and
V
=
X
X
+
Y
.
(i) Show that the joint density of (
X, Y
) is
f
(
x, y
) =
λ
α
+
β
Γ(
α
1
) Γ(
α
2
)
e

λ
(
x
+
y
)
x
α

1
y
β

1
,
x >
0
, y >
0
.
(ii) Compute the joint density of
U
and
V
.
Deduce that they are independent and
write down their marginal densities.
(c) Let
X
be a random variable with distribution function
F
(
x
).
Let
f
(
x
) be the
density function of
X
. Evaluate
R
∞
∞
F
(
x
)
f
(
x
)
dx
. (Hint: How is
F
(
X
) distributed?) ( 5 +
10 + 5 = 20 points)
(2) . (a) Consider two groups of patients, say Group A and Group B. The number of patients
in each of these groups is 50. Group A patients were on a blood pressure drug for 4 weeks
while Group B patients were on placebo.
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 Winter '08
 moulib
 Statistics, Standard Deviation, Variance, Probability theory, probability density function

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