Stat 341 – Homework 1, Due Jan 19
[12 pts] 1. Answer the following short questions:
(a) What is the value of
R ∞ 5 −y
y e dy?
0
(b) What is another name for the Gamma(1, 3) distribution?
(c) What is the constant c if f (x) = cx(1 − x)2 is a pdf?
(d) Are X and Y independent if their joint distribution is f (x, y) ∝
x, y ≤ 1?
(x2 −x)e−y
y!
for 0 ≤
[30 pts] 2. Briefly answer the following:
(a) Rewrite the following expression such that no complement appears.
P (A ∩ B) =?
(b) A particular toy includes 5 pieces of track. The pieces can be combined in any order
to make a distinct course. The toy comes with 10 racing cars. Each race consists
of 2 cars running on one course. How many distinct races can be run if all 5 track
pieces are used?
(c) Let Y have a Geometric distribution. If it costs a seconds per trial, the total time
to finish the first success is W = aY . Determine the number of seconds a and the
probability of success p if the moment generating function of W is
mW (t) =
0.2e3t
1 − 0.8e3t
(d) Suppose 70% of cars use lane 1 through an intersection, and the rest use lane 2. The
probability of an accident in the intersection is 1×10−5 . If the probability of getting
in an accident while in lane 2 is twice as high as in lane 1, find the probability of
an accident for a car passing through the intersecton in lane 1.
(e) If the time it takes to answer question i has distribution Xi ∼ Exponential(2),
there are 10 questions, and answer times are independent, what are the mean and
variance of the total time it takes to answer all 10 questions, Y = X1 + · · · + X10 ?
(f) Only 1 in 10 people you meet are smarter than you and the number of new people
you meet each year is a Poisson random variable with mean 100. If Y is the number
of people you meet in a year who are smarter than you, what is P (Y > 10)?
[8 pts] 3. (a) You place 20 $1 bills, 10 $10 bills, 2 $50, and 1 $100 bill in a hat and you ask a
random passerby to draw two bills without replacement from the hat. Why is it
incorrect to estimate the mean winnings you expect to pay as
10
2
1
20
+ 10 + 50 + 100
?
2×
33
33
33
33
(b) Continuing, use R to estimate the mean winnings you are expecting to pay.
[10 pts] 4. Let pair (X, Y ) be the midterm I and midterm II grades of a random student taking Stat
341. The random variables realized in this class are plotted in Figure 1. Suppose there
is an unknown joint distribution f (x, y), from which these observations are realizations.
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10
Midterm II Score
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0
10
20
30
40
Midterm I Score
Figure 1: Midterm I and II grades for this semester’s Stat 341 course.
(a) Supposing that the top score possible on both exams is 50 points, write an expression
for the covariance Cov(X, Y ) in terms of f (x, y) 1 .
(b) Circle the correlation coefficient most consistent with the plotted data in Fig. 1?
ρ = 0.001
ρ = −0.84
ρ = 0.80
(c) Draw on Fig. 1 an “eyeball” approximation to the marginal distribution f (x) based
on the observed data.2
(d) As y increases, does the mean of distribution f (x | y) (circle one)
increase
decrease
stay the same?
[20 pts] 5. Consider the following distribution for random polar coordinates (R, Θ) defined on the
unit circle.
3
(1 − r2 ), 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π.
f (r, θ) =
4π
(a) Find the marginal distributions, f (θ) and f (r), and name the marginal distribution
of θ.
(b) Find the expectations E[R] and E[Θ].
1
Write an expression with unsimplified, definite integrals and defining any functions or constants you use
in the expression and how they would be obtained from f (x, y).
2
When drawing, pretend the y-axis is the value f (x), but don’t worry about the magnitude of f (x), just
the shape of the marginal distribution.
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(c) What is Cov(R, Θ)?
(d) Suppose f (r, θ) represents the military’s targetting distribution. In other words, if
the target is at (0, 0), then the bomb lands at point (r, θ) with density f (r, θ). If
19
, what is E[C]?
the cost of missing a target is C = 300R2 + 500 and V (R) = 320
[20 pts] 6. You run a small garden center. Suppose the first spring weekend you stock 10 sugar
maples, 6 red oaks, and 4 Ohio buckeyes. The number of customers C wanting to buy
a tree during the weekend follows a Poisson distribution with mean λ = 12.
(a) The number of actual trees sold that weekend is
(
N = g(C) =
C, 0 ≤ C < 20
20, C ≥ 20.
Find the expected number E[N ] of trees sold. You will likely want to do the
calculation in R.
(b) If Y1 =# maples sold and Y2 =# oaks sold, use the counting method to derive the
probability P (Y1 = 4, Y2 = 3 | C = 10) that 4 maples and 3 oaks are purchased
when a total of 10 customers choose trees randomly?
(c) Using the result from part (b) above, generalized to arbitrary Y1 = y1 and Y2 =
y2 , show that the joint distribution p(y1 , y2 , c) of the trivariate random variable
(Y1 , Y2 , C) is
(y101 )(y62 )(c−y14−y2 ) (12)c −12
× c! e ,
0 ≤ c < 20
(20c )
p(y1 , y2 , c) = 10 6 4 × (12)c e−12 , c ≥ 20


y1
c!
y2
20−y1 −y2


0,
otherwise.





for 0 ≤ y1 ≤ 10, 0 ≤ y2 ≤ 6, and min{c, 16} − 4 ≤ y1 + y2 ≤ min{c, 16}.
(d) Use R and p(y1 , y2 , c) to compute the probability a total of N = C = 10 trees were
sold if Y1 = 4 and Y2 = 3? (Again, you’ll probably want R to help with calculations.)
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