The exam is composed by 7 independent exercises.

You must complete this exam on your own. If you need clarification on any question, you may ask one of the instructors.  You should state and sign the honor code below and turn in this page, along with your paper- work. Goodluck!

I have neither given nor received any unauthorized aid on this examination, nor have I concealed any violations of the honor code.

Exercise1. (5points) We throw simultaneously two fair dices. Compute the expectation of the sum of the square of each result.

Exercise 2. (10points) A bowl contains twenty cherries, exactly fifteen of which have had their stones removed. A greedy pig eats five whole cherries , picked at random, without remarking on the presence or absence of stones . Subsequently, a cherry is picked randomly from the remaining fifteen.

1.  (5points) What is the probability that this cherry contains a stone?

2.  (5points) Given that this cherry contains a stone, what is the probability that the pig consumed at least one stone?

Exercise 3. (10points)

1.  (5points) Let X be a random variable with a geometric distribution with parameter p e (0, 1). Let n be a positive integer. Prove that for any k > 0 we have P(X 一 n = klX > n) = P(X = k).

2.  (5points) Conversely, let Y a random variables with values in {1, 2, . . . }. Assume that P( Y 一 1 = klY > 1) = P( Y = k) for any k > 0. Prove that Y has a geometric distribution and specify the parameter.

Exercise 4. (15points) In an oil region,the probability that one drilling leads to an oil slick is p = 0.1. We made 10 oil drillings. Let X be the number of drillings leading to an oil slick.

1.  (5points) Under which assumptions X can be modelized with a Binomial law? Precise the parameters.

2.  (5points) Assume that X follows a Binomial law. Compute the probability that exactly two drillings lead to oil slicks.

3.  (5points) Assume that X follows a Binomial law. Compute the probability that at least one drilling lead to an oil slick.

1.  (5points) Find c such that f  is a density function

2.  (10points) Compute the expected weekly gasoline demand in thousands of gallons and its variance.

3.  (5points) The station is replenished every Monday at 8:00 pm. What must be the capacity of the gas tank so that the probability of running out of gas is less than 10−5 ?

Exercise 6. (25points)

We set f (x, y) = c exp (_2y)if y 三 x 三 0 and f (x, y) = 0 otherwise.

1.  (5points) Find c such that f  is a density on R2 .

From now on, we assume that the pair of random variables (X, Y) has the density f  for this choice of c.

2.  (5points) Find the marginal densities fX and fY of X and Y.

3.  (5points) Are X and Y independent?

4.  (10points) Find the density of X/Y.

Exercise 7.  (15points) We roll 10 fair dice simultaneously and independently.  Let X denotes the total of the result.

1.  (5points) Compute the expectation of X denoted by E [X] and the variance of X.

2.  (10points) Find the approximate probability that the sum obtained is between 30 and 40, inclusive.

Hint: please have a look to the normal table attached