xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

统计代写-S203031

时间：2021-01-24

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.