Prof. Elvezio Ronchetti Research Center for Statistics and GSEM
Ass : Linda Mhalla University of Geneva
S203031 Probability and Statistics II Fall 2017
EXAM January 22, 2018
– 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 (6 points)
A gambling game works as follows. A random variable X is produced ; you win 1 Fr if X > 0 and
you lose 1 Fr if X < 0. Suppose first that X has a normal (0, 1) distribution. Then the game is clearly
fair. Now suppose the casino gives you the following option. You can make X have a normal (b, 1)
distribution, but to do so you have to pay c · b Fr which is not returned to you even if you win. Here c > 0
is set by the casino, but you can choose any b > 0.
1. Defines the random variable G, your gain, as a function of X .
2. For what values of c is it advantageous for you to use this option ?
Exercise 2 (8 points)
Let X and Y have joint density
fX,Y (x, y) =
{
e−y if 0 < x < y <∞,
0 otherwise.
1. Compute Pr(Y > X + 1).
2. Give the marginal density of Y .
3. Give the conditional density of X|Y and compute the conditional expectation E(X|Y ).
4. Give the distribution of E(X|Y ).
Exercise 3 (16 points)
Let X1, . . . , Xn be an i.i.d. sample from a continuous distribution with cumulative distribution func-
tion
FX(x|θ) =
{
a + be−x
2/θ if x > 0,
0 otherwise,
where θ ∈ Θ = (0,∞) is an unknown parameter, a and b are some constants.
1. Show that a = 1 and b = −1.
2. Find fX(.|θ) the probability density function of X .
3. Show that Eθ(Xi) =
√
piθ/2 and use this fact to derive a method of moments estimator θˆMM of θ.
Hint : You can use the fact that
∫ +∞
−∞
1√
2piσ
x2e−x
2/(2σ2)dx = σ2, ∀σ2 > 0.
4. It can be shown that Eθ(X2i ) = θ. Using this fact, find Eθ(θˆMM).
Is θˆMM an unbiased estimator of θ ?
Is θˆMM a consistent estimator of θ ?
5. Let θ˜ = n[min(X1, . . . , Xn)]2. Find the cumulative distribution function of θ˜.
Hint : {min(X1, . . . , Xn) > u} if and only if {X1 > u} ∩ {X2 > u} ∩ · · · ∩ {Xn > u}.
6. Is θ˜ an unbiased estimator of θ ?
7. Find the log-likelihood function and show that the maximum likelihood estimator of θ is given by
θˆML =
1
n
n∑
i=1
X2i .
8. Compute the median of the distribution of Xi and give its maximum likelihood estimator.