MAN006 Assignment3

Indian Institute of Technology Roorkee
Spring Semester 2020-21

MAN-006: Probability and Statistics
Assignment 3
(Some standard discrete distributions)
(1) Let X be a random variable having a binomial distribution with parameters n = 25 and
p = 0.2. Evaluate P [X < µ − 2σ].
(2) Using Chebyshev’s inequality, show that the probability that in 900 flips of a balanced
35
coin the head will occur between 360 and 540 times is at least 36 .
(3) Suppose an unbiased die is cast at random seven times independently. What is the prob-
ability that each side appears at least once given that side 1 appears exactly twice?
(4) If X is a random variable such that E[X] = 10 and σX = 3, can X have a negative bino-
mial distribution N B(r; p) in which X represents the number of failures preceding the rth
success? Explain.
(5) If the probability is 0.75 that any person will believe a rumour, find the probabilities that
(i) the fifth person to hear it is the first to believe it (ii) the eighth person to hear the
rumour will be the fifth to believe it (iii) at least 4 persons do not believe the rumour
before the tenth person believes it.
(6) The probability of a successful rocket launching equals 0.8. Suppose that launching at-
tempts are made until 3 successful launchings have occurred.
(a) Find the probability that (i) exactly 6 attempts will be required (ii) fewer than 6
attempts will be required.
(b) Suppose that each launching costs Rs. 5 lakh. In addition, a launching failure results
in an additional cost of Rs. 50 thousand. Evaluate the expected cost for launchings
(for attempts until 3 successful launchings have occurred).
(7) Let X have a geometric distribution. Show that P (X > k + j|X > k) = P (X > j) (the
memoryless property of X), where k, j are non-negative integers.
(8) If a random variable X has a Poisson distribution such that P (X = 1) = P (X = 2), find
(i) P (X = 4), (ii) P (X ≥ 3).
(9) A merchant has found that the number of items of brand A that he can sell in a day is
a Poisson random variable with mean 4. What is the expected number of days out of 25
that the merchant will sell no items of brand A?
(10) There are 270 typographical errors in a book of 675 pages, and these errors are randomly
distributed throughout the book. Find the probabilities that (i) a randomly chosen page
will have more than two errors (ii) exactly three of the five randomly chosen pages will be
free of errors (iii) five randomly chosen pages will have less than three errors altogether.
(11) Let X have a Poisson distribution. If P (X = 1) = P (X = 3), find the mode of the
distribution.
(12) Show that if the limiting condition n → ∞, p → 0, while np remains constant are applied
to the moment generating function (mgf) of the binomial distribution, we get the mgf of
1
2
a Poisson distribution.
(13) Suppose that a container contains 10, 000 particles. The probability that such a particle
escapes from the container is equal to 0.0004. What is the probability that more than 5
such escapes occur?
(14) A radioactive source is observed during 7 time intervals of 10 seconds each. If the number
of particles emitted during each time interval follows a Poisson probability law with a rate
of 0.5 particles per second, what is the probability that there are at least two time intervals
during each of which no more than 2 particles are emitted.
(15) A machinists keeps a large number of washers in a drawer. About 50% of these washers
are 0.5 inch in diameter, about 30% are 1 inch in diameter, and about 20% are 1.5 inch
in diameter. Suppose that 10 washers are chosen at random. Find the probabilities that,
among the chosen ones, there are
(a) exactly five 0.5-inch washers, three 1-inch washers and two 1.5-inch washers .
(b) all three kinds of washers.
(c) three of one kind, three of another kind and four of the third kind.
9
(16) If the mgf of a random variable X is 32 + 13 et , determine P (|X − µ| < 2σ).
t
(17) If the mgf of a random variable X is e100(e −1) , find a lower bound on P (75 < X < 125)
using Chebyshev’s inequality.
(18) Let X denote the number of successes throughout n independent repetitions of a random
experiment having probability of success 41 . Determine the smallest value of n so that
probability of at least one success is at least 0.80.
(19) Let X ∼ b(n, p) (a binomial random variable with parameters n and p). Using Chebyshev’s
inequality, show that for any > 0,

X
lim P − p < = 1 .
n→∞ n
(20) Let X ∼ b(n, p). If x = r is the unique mode of X, show that

(n + 1)p − 1 < r < (n + 1)p .
(21) Let X be a random variable with mean µ and variance σ 2 such that the third moment
E[(X − µ)3 ] about the vertical line through µ exists. The value of the ratio E[(X − µ)3 ]/σ 3
is often used as a measure of skewness. Graph each of the following probability density
functions and show that this measure is negative, zero and positive for these respective
distributions (which is said to be skewed to the left, not skewed, and skewed to the right,
respectively).
(a) f (x) = (x + 1)/2, −1 < x < 1, zero elsewhere.
(b) f (x) = 12 , −1 < x < 1, zero elsewhere.
(c) f (x) = (1 − x)/2, −1 < x < 1, zero elsewhere.
(22) Let X be random variable with mean µ and variance σ 2 such that the fourth moment
E[(X − µ)4 ] about the vertical line through µ exists. The value of the ratio E[(X − µ)4 ]/σ 4
is often used as a measure of kurtosis. Graph each of the following probability density
functions and show that this measure is smaller for the first distribution.
(a) f (x) = 12 , −1 < x < 1, zero elsewhere.
(b) f (x) = 3(1 − x2 )/4, −1 < x < 1, zero elsewhere.
3
ANSWERS
(1) (0.8)25 (2) 35

36
(3) 24
625
(4) No
(5) (i) (0.25)4 (0.75) (ii) 35 (0.75)5 (0.25)3

P6 9
(0.75)r (0.25)10−r

(iii) r=1 r−1
(6) (a) (i) 0.041 (ii) 0.94208 (b) 19.125 lakh
(8) (i) 23 e−2 (ii) 1 − 5e−2 (9) 0.458 (approx.)
(10) (i) 1 − (1.48)e−0.4 (ii) 10e−1.2 (1 − e−0.4 )2 (iii) 5e−2
(11) 2 (13) 0.215 (14) 1 − 2 (0.875)7 (approx.)
10!
(0.5)x1 (0.3)x2 (0.2)x3 ,
P
(15) (a) 0.085 (b) x1 !x2 !x3 !
x1 , x2 , x3 ≥ 1, x1 + x2 + x3 = 10
10!
(c) 3!3!4!
[(0.5)3 (0.3)3 (0.2)4 + (0.5)3 (0.3)4 (0.2)3 + (0.5)4 (0.3)3 (0.2)3 ]
5
9 1 x 2 9−x
P
(16) x 3 3
(17) 0.84 (18) 6
x=1

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Indian Institute of Technology Roorkee

Spring Semester 2020-21

(20) Let X ∼ b(n, p). If x = r is the unique mode of X, show that

(1) (0.8)25 (2) 35

(5) (i) (0.25)4 (0.75) (ii) 35 (0.75)5 (0.25)3

(6) (a) (i) 0.041 (ii) 0.94208 (b) 19.125 lakh

(8) (i) 23 e−2 (ii) 1 − 5e−2 (9) 0.458 (approx.)

(10) (i) 1 − (1.48)e−0.4 (ii) 10e−1.2 (1 − e−0.4 )2 (iii) 5e−2

(11) 2 (13) 0.215 (14) 1 − 2 (0.875)7 (approx.)

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