Download 2nd sheet : discrete random variables

L2-PC-C
Probabilités - Statistiques
2011/2012
2nd sheet : discrete random variables
Exercise 1 : We say that a real random variable X follows a uniform distribution on {1, ..., n} when
the probability that X takes a value of {1, ..., n} is constant.
1. Describe X(Ω)
2. Find the constant for which X is a probability distribution.
3. Find the expectation and the variance of X .
Exercise 2 : The number X of kilos of tomatos harvested in a garden during a week is a random
variable whose probability distribution is the following
x
P {X = x}
0
1
2
3
0,1 0,5 0,3 0,1
1. Which is the expected value of X ? Which is its variance ?
2. During the 6 weeks of the harvest season, the probability distribution does not change. Find the
expectation and the variance for the random variable Y which describes the total harvest in the
6 weeks. We are assuming here that the 6 harvest week are independent.
Exercise 3 : A lottery has 1000 tickets. One among these wins 500 euros. Other two tickets win 100
euros. Fifty other tickets win 10 euros. What should be the ticket prize for this lottery to be a fair
game ?
Exercise 4 : We consider two planes, a twin-engined jet T and a four-engined jet F. We assume that
each engine of these planes has the same chance p of breaking down, and that the engines are mutually
independent. Let X be the random variable which counts the number of engines of T breaking down
and Y be the random variable which counts the number of engines of F breaking down.
1. Find the probability distributions of X and Y respectively.
2. It is known that a plane can accomplish its ight only if at least half of its engines work. Let
PT and PF be the probability for a ight to be accomplished by a twin-engined jet and by a
four-engined jet respectively. Calculate PT and PF depending on p. According to the value of p,
say which plane is more safe.
Exercise 5 : It is known that the screws made in a factory have a probability of 0,01 of being faulty.
The state of each screw is independent of the state of the other screws. The enterprise accepts to refund
any sold 10 screws package provided that more then one of the contained screws are faulty. In which
rate the sold packages have to be refunded ?
Exercise 6 : A light bulb has a probability of 0,2 of lasting more then 2 years. On a ceiling there
are 5 bulbs. Compute the probability
1. of not changing any bulb during the 2 rst years
2. of changing all the bulbs during the 2 rst years
Exercise 7 : 2 women and 3 men stand on a platform at the railway station. 2 persons are randomly
chosen for a survey. Let X be the number of chosen women.
1. Give the probability distribution of X
2. Which is the probability of choosing at least a women ?
3. Calculate E(X) and σ(X).
Exercise 8 : An urn contain N white marbles and M red marbles. We draw the marbles one by one
with replacement untill a red marble is drawn.
1. Which is the probability that exactly n draws are needed ?
2. Which is the probability that al least n draws are needed ?
Exercise 9 : We play "heads or tails" with an unfair coin. We let p be the probability of obtaining
"heads". We toss the coin until a rst "head" appear, and we set X the number of the coin toss at
which we obtained the rst "head".
1. Give the probability distribution of X , and check that this is a well-dened probability distribution.
(
X
geometric
p)
2. Compute the expectation and the variance of X
3. Let X1 and X2 be independent random variables following geometric distributions with paramenter p1 and p2 respectively. Give the distribution of X1 + X2 , as well as its expectation and
variance.
We say that follows a
distribution with parameter
Exercise 10 : An electrician buys some components in packages containing 10 items each one. He
always checks randomly 3 components in each package, and accepts a package only if all the 3 checked
items are without imperfections. If 30% of the packages contain 4 faulty items and the remaining 70%
contain only 1 faulty item, in which rate will the packages be accepted ?
Exercise 11 : Let X and Y be independent random variables following Poisson distributions with
paramenter λ and µ respectively.
1. Give the distribution of X + Y .
2. Calculate E
1
1+X
Exercise 12 : In order to check a telematic network, some tests are done to verify the connection to
a central unity, and it is discovered that 95% of the attempts give good connection. A society has to
connect 4 times a day to upload its les. Let X be the number of attempts needed to get connection
4 times.
1. Calculate P (X = 4).
2. Calculate the probability that more then 6 attempts are needed.
3. Calculate E(X) and σ(X).
Exercise 13 : The number of cold caught by a given person during a year is represented by a random
variable following a Poisson distribution of parameter λ = 5. Assume that a miracolous cure has brought
out so that the parameter λ has decreased to 3 for 25% of the population. For the remaining 75% the
cure does not work at all. A person tests this cure during 1 year and catches two colds. Which is the
probability that this cure works well for him ?