Department of Mathematics, IIT MadrasMA 2040 : Probability, Statistics and Stochastic Processes

January - May 2015

Problem Set - V

13.03.2015

1. Each of n packages is loaded independently onto either a red truck (with probability p) or onto a green

truck (with probability 1 p). Let R be the total number of items selected for the red truck and letG be the total number of items selected for the green truck.

(a) Determine the PMF, expected value, and variance of the random variable R.

(b) Evaluate the probability that the first item to be loaded ends up being the only one on its truck.

(c) Evaluate the probability that at least one truck ends up with a total of exactly one package.

(d) Evaluate the expected value and the variance of the difference RG.(e) Assume that n 2. Given that both of the first two packages to be loaded go onto the red truck,

find the conditional expectation, variance and PMF of the random variable R.

2. Gopal fails quizzes with probability 1/4, independent of other quizzes.

(a) What is the probability that Gopal fails exactly two of the next six quizzes?

(b) What is the expected number of quizzes that Gopal will pass before he has failed three times?

(c) What is the probability that the second and third time Gopal fails a quiz will occur when he takes

his eighth and ninth quizzes, respectively?

(d) What is the probability that Gopal fails two quizzes in a row before he passes two quizzes in a

row?

3. A computer system carries out tasks submitted by two users. Time is divided into slots. A slot can

be idle, with probability PI = 1/6, and busy with probability PB = 5/6. During a busy slot, there is

probability P1|B = 2/5 (respectively, P2|B = 3/5) that a task from user 1 (respectively, 2) is executed.We assume that events related to different slots are independent.

(a) Find the probability that a task from user 1 is executed for the first time during the 4th slot.

(b) Given that exactly 5 out of the first 10 slots were idle, find the probability that the 6th idle slot

is slot 12.

(c) Find the expected number of slots up to and including the 5th task from user 1.

(d) Find the expected number of busy slots up to and including the 5th task from user 1.

(e) Find the PMF, mean, and variance of the number of tasks from user 2 until the time of the 5th

task from user 1.

4. Consider a Bernoulli process with probability of success in each trial equal to p.

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(a) Relate the number of failures before the rth success (sometimes called a negative binomial random

variable) to a Pascal random variable and derive its PMF.

(b) Find the expected value and variance of the number of failures before the rth success.

(c) Obtain an expression for the probability that the ith failure occurs before the rth success.

5. A train bridge is constructed across a wide river. Trains arrive at the bridge according to a Poisson

process of rate = 3 per day.

(a) If a train arrives on day 0, find the probability that there will be no trains on days 1, 2, and 3.

(b) Find the probability that the next train to arrive after the first train on day 0, takes more than

3 days to arrive.

(c) Find the probability that no trains arrive in the first 2 days, but 4 trains arrive on the 4th day.

(d) Find the probability that it takes more than 2 days for the 5th train to arrive at the bridge.

6. Type A, B, and C items are placed in a common buffer, each type arriving as part of an independent

Poisson process with average arrival rates, respectively, of a, b, and c items per minute. Assume the

buffer is discharged immediately whenever it contains a total of ten items.

(a) What is the probability that, of the first ten items to arrive at the buffer, only the first and one

other are type A?

(b) What is the probability that any particular discharge of the buffer contains five times as many

type A items as type B items?

(c) Determine the PDF, expectation, and variance for the total time between consecutive discharges

of the buffer.

(d) Determine the probability that exactly two of each of the three item types arrive at the buffer

input during any particular five minute interval.

7. A store opens at t = 0 and potential customers arrive in a Poisson manner at an average arrival rate

of potential customers per hour. As long as the store is open, and independently of all other events,

each particular potential customer becomes an actual customer with probability p. The store closes as

soon as ten actual customers have arrived.

(a) What is the probability that exactly three of the first five potential customers become actual

customers?

(b) What is the probability that the fifth potential customer to arrive becomes the third actual

customer?

(c) What is the PDF and expected value for L, the duration of the interval from store opening to

store closing?

(d) Given only that exactly three of the first five potential customers became actual customers, what

is the conditional expected value of the total time the store is open?

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