Semester 2, Session 2012/2013

ASSIGNMENT (10%)

Instructions:

a. This is a GROUP assignment. b. Each student must be a member of a group of 4 or 5 students, selected by lecturer. c. Solutions from each group must be submitted by 19 April 2013.

SPECIAL DISTRIBUTIONS

I. Concept of probability (3%)

1. Explain why the distribution B(n,p) can be approximated by Poisson distribution with parameter if n tends to infinity, p 0, and = np can be considered constant.

2. Show that – and + are the turning points in the graph of the p.d.f. of normal distribution with mean and standard deviation .

3. What is the relationship between exponential distribution and Poisson distribution?

II. Computation of probability (7%)

1. Let the random variable X follow a Binomial distribution with parameters n and p. We write X ~ B(n,p). * Write down all basic assumptions of Binomial distribution. * Knowing the p.m.f. of X, show that the mean and variance of X are = np, and 2 = np(1 – p), respectively.

2. A batch contains 40 bacteria cells and 12 of them are not capable of cellular replication. Suppose you examine 3 bacteria cells selected at random without replacement. What is the probability that at least one of the selected cells cannot replicate?

3. Redo problem No. 2 if the 3 bacteria cells are selected at random with replacement.

4. The number of customers who enter a bank in an hour follows a Poisson distribution. If P(X = 0) = 0.05, determine the mean and variance of the number of customers in an hour.

5. In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. What is the probability that there are no log-ons in an interval of 6 minutes?

6. The time until recharge for a battery in a laptop computer under common conditions is normally distributed with