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Sample Problems for Exam 3
STA 4321
Mathematical Statistics I
Fall 1999
These questions are only meant as a study aid and to help you test your knowledge. Being able to solve them does not
guarantee that you are well-prepared for the exam.
1. For each of the following joint densities, indicate whether Y1 and Y2 are independent (Yes or No). No explanation is required.
(a)
f (y1 , y2 ) =
2, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, 0 ≤ y1 + y2 ≤ 1,
0, elsewhere.
f (y1 , y2 ) =
(b)
(c)
f (y1 , y2 ) =
e−(y1 +y2 ) , y1 ≥ 0, y2 ≥ 0,
0,
elsewhere.
y1 + y2 , 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1,
0,
elsewhere.
(d)
f (y1 , y2 ) =
e−y1 , y1 ≥ 0, 0 ≤ y2 ≤ 1,
0,
elsewhere.
(e)
f (y1 , y2 ) =
e−y1 , 0 ≤ y2 ≤ y1 < ∞,
0,
elsewhere.
2. Let Y1 and Y2 denote the proportion of time, out of one workday, that employees I and II, respectively, actually spend
performing their assigned tasks. The joint relative frequency behavior of Y1 and Y2 is modeled by the density function
y1 + y2 , 0 ≤ y1 ≤ 1; 0 ≤ y2 ≤ 1
f (y1 , y2 ) =
0,
elsewhere.
(a) Find P (Y1 ≥ 12 , Y2 ≥ 12 ).
(b) Find the marginal density function for Y2 .
(c) Find P (Y2 ≥ 12 ).
(d) Find P (Y1 ≥ 12 |Y2 ≥ 12 ).
(e) Find the conditional density of Y1 given Y2 .
(f) Find P (Y1 ≥ 21 |Y2 = 14 ).
3. Suppose that the random variables Y1 and Y2 have joint density given by
(
1
+ 2y1 y2 , 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1,
f (y1 , y2 ) = 2
0,
elsewhere.
Note that f (y1 , y2 ) is symmetric in y1 and y2 , so that Y1 and Y2 have the same marginal distributions, and hence the same
mean and variance.
(a) Find E(Y1 ).
(b) Find V (Y1 ).
(c) Find Cov(Y1 , Y2 ).
(d) Find E(2Y1 + 2Y2 + 5).
(e) Find V (2Y1 + 2Y2 + 5).
(f) Find Cov(2Y1 − Y2 , Y1 + Y2 ).
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STA 4321 Sample Problems for Exam 3
4. Suppose that V (Y1 ) = V (Y2 ) = σ 2 and Cov(Y1 , Y2 ) = γ. Find
(a) V (Y1 + Y2 )
(b) V (Y1 − Y2 )
(c) Cov(Y1 + Y2 , Y1 − Y2 )
5. Suppose that Z ∼ N (0, 1) and X ∼ χ2ν are independent, and let
Z
T =p
X/ν
.
Find E(T ) and V (T ). What must you assume about ν?
6. Suppose that X1 ∼ χ2ν1 and X2 ∼ χ2ν2 are independent, and let
F =
X1 /ν1
.
X2 /ν2
Find E(F ) and V (F ). What must you assume about ν1 or ν2 ?
7. Let the random variable Y have density
f (y) =
(
1
2 (1
+ y), −1 ≤ y ≤ 1,
0,
elsewhere.
Find the density of W = Y 2 .
8. Let Y1 and Y2 have joint density
(
3y1 , 0 ≤ y2 ≤ y1 ≤ 1,
f (y1 , y2 ) =
0,
elsewhere,
and let W = Y1 − Y2 . Find the density of W .
9. The Romulans have trapped the starship Enterprise in a spherical energy bubble. Spock has determined that at any given time,
the radius of the bubble, Y , is a random variable with probability density function
(
72y 2 /(9 + 16π 2 y 6 ), y > 0,
fY (y) =
0,
elsewhere.
In order to plan an escape, Captain Kirk desperately needs to know the probability density function of the volume, V , of the
the bubble. Knowing that the volume of a sphere of radius Y is given by the formula
V =
4
πY 3 ,
3
Spock uses the method of transformations to find the probability density function of V . Since Captain Kirk’s algebra is not
very good, Spock also simplifies his result. What is Spock’s answer?
10. Suppose that Y1 , . . . , Y5 are independent, exponentially distributed random variables, each with mean β = 10. Use the
method of moment generating functions, or results derived in class, to find the density of
Y =
1
(Y1 + Y2 + Y3 + Y4 + Y5 ) .
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STA 4321 Sample Problems for Exam 3
11. Let Y1 and Y2 be independent Poisson random variables with means λ1 and λ2 , respectively.
(a) Use the method of moment generating functions to show that the distribution of W = Y1 + Y2 is Poisson with mean
λ 1 + λ2 .
(b) Use the result of part (a) to show that the conditional probability function of Y1 given W = w is binomial with n = w
and p = λ1 /(λ1 + λ2 ). Hint: Pr(Y1 = y1 , W = w) = Pr(Y1 = y1 , Y2 = w − y1 ).
12. Let Y1 , . . . , Y10 be a random sample from the distribution with density function
2y/θ2 , 0 ≤ y ≤ θ,
f (y) =
0,
elsewhere,
where θ > 0.
(a) Find the density of X = max(Y1 , . . . , Y10 ).
(b) Find E(X) and Var(X).
13. Consider the probability density function
(
0,
y < 0,
f (y) =
−2
(1 + y) , y ≥ 0.
Suppose that U ∼ U (0, 1). Find a transformation h(U ) so that Y = h(U ) has density f .
14. Short answer. You are not required to show any work.
(a) Suppose that Y1 , Y2 , and Y3 are independent exponential random variables, each having mean 10, and let X = Y1 +
Y2 + Y3 . Name the distribution of X and give the values of any parameters.
(b) Refer to part (b). For what value of c does W = cX have a chi-square distribution? With how many degrees of freedom?
15. Short answer. You are not required to show any work.
Suppose that Y1 and Y2 are independent normal random variables each with mean µ and variance σ 2 . Name the distribution
of each of the following random variables and give the values of any parameters.
(a) X = Y1 + Y2
2 2
Y1 − µ
Y2 − µ
+
(b) W =
σ
σ
2
Y1 + Y2 − 2µ
√
(c) V =
Hint: Refer to part (a).
2σ 2
16. Let Y ∼ Beta(1, θ), so that Y has probability density function
f (y) = θ(1 − y)θ−1 ,
0 < y < 1.
(a) Show that X = − ln(1 − Y ) ∼ Exp(β = 1/θ).
(b) Now suppose that
PnY1 , . . . , Yn are independent random variables, each having a Beta(1, θ) distribution. Name the distribution of U = i=1 [− ln(1 − Yi )] and give the values of any parameters.
(c) In the situation of part (b), find E(1/U ).