## STAT2001 – Week 2 Assignment Paper.

STAT2001 – Week 2 Assignment Paper.

__Do all 8 questions. Show your steps clearly__.

Deadline for this assignment is * 17^{th} Oct. 6:30p.m.* You can submit to the assignment locker (next to LSB 125) or to your Tutors.

- (8 marks) An internet service provider uses 50 modems to serve the needs of 1000 customers. STAT2001 – Week 2 Assignment Paper.It is estimated that at a given time, each customer will need a connection with probability 0.01, independently of the other customers. What is the probability mass function (p.m.f.) of the number of modems in use at the given time?

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- Let
*X*be a random variable with p.m.f.:

(a) (5 marks) Find *a.*

(b) (5 marks) Find *E(X).*

(c) (5 marks) Find *Var(X)*.

- (10 marks) Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has 10 white and 10 black balls. Find the probability that 3 white balls are drawn before 2 black balls. STAT2001 – Week 2 Assignment Paper.

- (10 marks) The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter λ. However, such a random variable can be observed only if it is positive, since if it is 0, then we cannot know that such an insect was on the leaf. If we let
*Y*denote the observed number of eggs, then

*P(Y=i)=P(X=i|X>0)*

where *X* is a Poisson random variable with parameter λ. Find *E(Y).*

- (10 marks) An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly 10 selections?

- A store selling newspapers orders only
*n=4*of a certain newspaper because the manager does not get many calls for that publication. If the number of requests per day follows a Poisson distribution with mean 3,

(a) (8 marks) What is the expected value of the number sold?

(b) (5 marks) How many should the manager order so that the chance of running out is less than 0.05?

- X is a Poisson random variable with mean λ:

(a)(10 marks) Use moment generating function, find the third moment of X.

(b)(10 marks) Use part (a), prove that

(Remark: this quantity is a measure of skewness. Note that it is always positive for Poisson. It is consistent with its right-skewed shape.)

P.T.O.

- Let
*X*be a random variable with probability mass function:

- (5 marks) Find the value
*C*. - (9 marks) Find the moment generating function of
*X*. STAT2001 – Week 2 Assignment Paper.