Prof. Elvezio Ronchetti Research Center for Statistics and GSEM
Ass : Julien Bodelet University of Geneva
S203031 Probability and Statistics II Fall 2019
EXAM January 17, 2020
— The exam lasts 3h.
— You may answer in English or in French.
— You are not allowed to leave the room during the exam.
— Open book exam.
— Justify your answers and do not forget to write your name on your copy.
Good luck !
Exercise 1 ( 11 points)
Let X1, X2, . . . , Xn be n i.i.d random variables representing the prices set by n competitors for a
given product and assume that Xi follows an exponential distribution with parameter λ, i.e. with density
fX(x) =
{
λe−λx if x > 0
0 otherwise ,
where λ is a positive unknown parameter. We want to investigate the statistical properties of the lowest
price in the market for the product, Y = min(X1, X2, ..., Xn).
1. Compute the cumulative distribution function FY (y) of Y .
2. Show that the density of Y is exponential with parameter nλ.
3. Compute the expected value and the interquartile range of Y .
4. We simulate M = 106 samples of size n = 10 of prices issued from an exponential distribution
with parameter λ = 1. For each sample we take the lowest price. What is the approximate average
value of the M lowest prices? What is the approximate percentage among the M lowest prices
which are larger than 0.1386? Justify your answers.
Exercise 2 ( 7 points)
LetX1, X2 be two independent random variables representing the number of insurance claims during
a month for two groups of drivers. We assume that Xi follows a Poisson distribution with parameter
λi, i = 1, 2.
1. Show that for every n ≥ 1, the conditional distribution of X1, given X1 + X2 = n, is binomial
and give the parameters of this binomial distribution.
2. Assume that the expected number of claims in each group is the same, i.e. E[X1] = E[X2]. If
the total number of claims for the two groups is 150, what is the probability that the number of
claims for the first group is at least 90?
Hint : Use the normal approximation.
Exercise 3 ( 12 points)
Let X1, X2, . . . , Xn be i.i.d random variables from a continuous distribution with density
fα(x) = α(α + 1)x
α−1(1− x) ,
where α > 0 and 0 < x < 1.
1. Show that µ = E[Xi] = αα+2 .
2. Show that αˆn = g(X¯n) = 2X¯n/(1 − X¯n) is the Method of Moments estimator of α. Is this
estimator consistent for α?
3. Let σ2 = var[X] = 2α
(α+2)2(α+3)
. Give the distribution of
√
n(X¯n − µ), when n→∞.
4. Show that
√
n(g(X¯n)− g(µ))→ N
(
0,
(
dg(µ)
dµ
)2
σ2
)
,
when n→∞.
Hint : Obtain an approximation of g(X¯n) by means of a first order Taylor approximation around
µ.
5. Using point 4., show that the distribution of
√
n(αˆn − α) isN (0, V (α)), when n→∞ and give
the function V (α).
6. We would like to use the estimator αˆn for testing. Write the the corresponding Studentized sta-
tistic and using point 5. give its distribution, when n→∞.