STAT 463/853 – Fall 2021
Midterm test 1
1. Name the pdf file of your solution as follows:
midterm 1+your name+your student#
2. Submit it by e-mail, by noon Thursday, October 28,
to the following e-mail address: [email protected]
3. Add the subject: midterm 1+your name+your student#
Thus, your name and student # should appear twice!
4. If you experience any difficulty trying to send your file,
due to its large size, send it in the zipped form!
In answering the following questions provide as complete explanation
as possible.
1. (Distribution of random variables) Let X ∼ U [0, 3] have uniform distri-
bution on the interval [0, 3]. What is the probability that X2 − 5X + 6 > 0?
2. (PDF and CDF) a) Consider the following family of probability densities
f(y) = t(p)py(1− p)1−y, y ∈ [0, 1], 0 < p < 1, (1)
defined on the interval [0, 1] and depending on the parameter p ∈ (0, 1). Here
t(p) is a normalizing coefficient. Determine the value of this coefficient.
b) (For students of the class STAT-853 only) Determine the cumulative
distribution function F (y) corresponding to the probability density (1).
3. (Expectation) A box contains n tickets, numbered 1, 2, ..., n. A ticket is
drawn at random from the box, its number is registered, and the ticket is
returned to the box. Another ticket is drawn at random from the box, again
its number is registered, and the ticket is returned to the box. This procedure
is repeated until first such moment T when all n numbers have been registered.
Obviously, the moment T can be represented as the following sum
T = T1 + T2 + T2 + · · ·+ Tn,
where T1 = 1 is waiting time till the first “new” number is obtained; T2 is the
waiting time till another “new” number is obtained, i.e., a number different
from the previous one; T3 is the waiting time till yet another “new” number
is obtained, different from the previous two; etc.
a) What is the distribution of i) T2 ? ii) T3 ? iii) Tn ?
b) Find the expected value ET .
1
4. (Moment generating function) The moment generating function (MGF)
of a random variable X is given by
MX(t) = E e
tX .
Suppose X is a gamma random variable whose pdf is given by
f(x) =

λα
Γ(α)
xα−1e−λx , x > 0,
0 , otherwise,
for some α > 0 and λ > 0.
a) Calculate the MGF MX(t).
b) For which t is MX(t) finite?
5. (Poisson, exponential, and geometric distributions) Suppose X has
exponential distribution E(λ). Denote P(λ) the Poisson distribution with
parameter λ > 0. Let Y be a random variable whose conditional distribution
given X = x is P(x). Thus, the joint distribution of X,Y is of the mixed type.
a) Determine the marginal distribution of Y .Hint: Recall the gamma integral,∫ ∞
0
xα−1e−λx dx =
Γ(α)
λα
.
b) Show that the random variable Y + 1 has a geometric distribution.
6. (LLN) Let X1, X2... be a sequence of independent, identically distributed
random variables having finite second moment and positive expected value
µ > 0. Prove that
lim
n→∞
P(X1 +X2 + · · ·+Xn > 100) = 1.
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