hw1

Course: ESO-209

Home Work: 1

Instructor: Debasis Kundu

1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times,

(d) an infinite number of times.

2. A coin is tossed until for the first time the same result appear twice in succession. To

an outcome requiring n tosses assign a probability 2−n. Describe the sample space.

Evaluate the probability of the following events:

(a) A = The experiment ends before the sixth toss.

(b) B = An even number of tosses are required

(c) A ∪ B, A ∩ B, A ∩ B′, A′ ∩ B′, A′ ∩ B

3. Consider the sample space S = {0, 1, 2, 3, ...}. Consider the sigma field of all subsets

of S. To the elementary event {j} assign the probability

P ({j}) = c2j

j!, j = 0, 1, ...

(a) Determine the constant c.

(b) Define the events A, B and C by

A = {j : 2 ≤ j ≤ 4}, B = {j; j ≥ 3}, C = {j; j is an odd integer}

Evaluate P (A), P (B), P (C), P (A∩B), P (A∩C), P (B∩C), P (A∩B∩C) and verify

the formula P (A ∪B ∪ C).

4. Let A and B be two arbitrary subsets of a sample space S. Find the smallest σ - field

generated by the class {A,B}. Let S = {1, 2, 3, 4, , 5, 6} represent the sample space by

tossing a die once and let A = {1, 3, 5}, B = {1, 4}. Specify the σ - field A generated

by C = {A,B}. For P (A) = 0.47, P (B) = 0.33, P (A∩B) = 0.17, assign probabilities

to all events to A.

5. Three tickets are drawn randomly without replacement from a set of tickets numbered

1 to 100. Show that the probability that the number of selected tickets are in (i)

arithmetic progression is 166

and (ii) geometric progression is 105

(1003 )

6. Three players A, B and C play a series of games, none of which can be drawn and

their probability of winning any game are equal. The winner of each game scores 1

point and the series is won by the player who first scores 4 points. Out of first three

games A wins 2 games and B wins 1 game. What is the probability that C will win

the series.

7. Thirteen cards are selected randomly without replacement from a deck of 52 cards.

Find the probability that there are at least 3 Aces given that there are at least 2 Aces

in the selected cards.

8. Urn I contains 3 black and 5 red balls and urn II contains 4 black and 3 red balls.

One urn is chosen randomly and a ball is drawn randomly which is red. Find the

probability that urn I was chosen.

9. A point P is randomly placed in a square with side of 1 cm. Find the probability that

the distance from P to the nearest side does not exceed x cm.

10. If six dice are rolled find the probability that at least two faces are equal.

Course: ESO-209

Home Work: 2


1. Find the probability pr that in a sample of r random digits no two are equal.

2. What is the probability that among k random digits (a) 0 does not appear, (b) 1 does

not appear, (c) neither 0 nor 1 appears, (d) at least one of the two digits 0 and 1 does

not appear? Let A and B represent the events in (a) and (b). Express the other events

in terms of A and B.

3. Find the value of N ≥ n such that the following probability is maximum (assume

n ≥ k),

P (N) =

(

k

i

)(

N−kn−i

)

(

N

n

)

4. Suppose that in answering a question on a multiple choice test an examinee either knows

the answer or he/she guesses. Let p be the probability that he will know the answer

and let 1 − p be the probability that he/she will guess. Assume that the probability

of answering a question correctly is 1 for an examinee who knows the answer and 1m

for an examinee who guesses; here m is the multiple choice alternatives. Find the

conditional probability that an examinee knew the answer to a question, given that he

has correctly answered it.

5. Thirteen cards are selected randomly without replacement from a deck of 52 cards.

Find the probability that there are at least 3 aces given that there are at least 2 aces

in the selected card.

6. Consider an urn in which 4 balls have been placed by the following scheme. A fair

coin is tossed; if the coin falls heads, a white ball is placed in the urn, and if the coin

falls tails, a red ball is placed in the urn. (a) What is the probability that the urn will

contain exactly 3 white balls? (b) What is the probability that the urn will contain

exactly three white balls given that the first ball placed in the urn was white?

7. A = {Ai; i ∈ Λ}, be a class of events. These events are said to be pairwise independent

if P (Ai1 ∩ Ai2) = P (Ai1)P (Ai2) for any two distinct elements Ai1 ∈ Λ and Ai2 ∈ Λ.

Remember two events A and B are called independent events if P (A∩B) = P (A)P (B).

Does pairwise independence imply independence?

8. Let A and B be two events. Assume P (A) > 0 and P (B) > 0. Prove that (i) if A and

B are mutually exclusive (A∩B = φ) then A and B are not independent (ii) if A and

B are independent then A and B are not mutually exclusive.

9. Let A, B and C be independent events. In terms of P (A), P (B) and P (C), express,

for k = 0, 1, 2, 3, (i) P(exactly k of the events A, B, C will occur), (ii) P(at least k of

the events A, B, C will occur), (iii) P (at most k of the events A, B, C will occur).

10. Let A and B be two independent events such that P (A ∩B) = 16. (i) If P ( neither of

A and B occurs) = 13, find P (A) and P (B), (ii) if P( A occurs and B does not occur)

= 13, find P (A) and P (B). For either part (i) and (ii), are P (A) and P (B) uniquely

determined?

Course: ESO-209

Home Work: 3


1. Show that if P (A ∩B ∩ C ∩D) > 0, then

P (A ∩B ∩ C ∩D) = P (A).P (B|A).P (C|A ∩B).P (D|A ∩B ∩ C).

In how many different ways you can express this probability? Generalize for n events.

2. If A and B are independent events, show that A and B are also independent events.

3. Two fair dice labeled I and II are thrown simultaneously and outcomes of the top faces

are observed. Let

A = Event that die I shows an even number

B = Event that die II shows an odd number

C = Sum of the two faces are odd

Are A, B and C independent events?

4. Let S = {HH, HT, TH, TT} and F = class of all subsets of S. Define X by X(ω) =

number of H’s in ω. Show that X is a random variable.

5. Let X be a random variable. Which of the following are random variables (a) X 2, (b)

1X

given that {X = 0} = φ, (c) |X|, (d)√X, given that {X < 0} = φ

6. Let S = [0, 1] and F be the Borel σ - field of subsets of S. Define X on S as follows:

X(ω) = ω if 0 ≤ ω ≤ 12and X(ω) = ω − 1

2if 1

2< ω ≤ 1. Is X a random variable?

Course: ESO-209

Home Work: 4


1. Verify, whether or not the following functions can serve as p.m.f

(a) f(x) = (x−2)2

for x = 1, 2, 3, 4.

(b) f(x) = e−λλx

x!for x = 1, 2, 3, . . ., (i) λ > 0, (ii) λ < 0.

2. A battery cell is labeled as good if it works for at least 300 days in a clock, otherwise

it is labeled as bad. Three manufacturers, A, B and C make cells with probability of

making good cell as 0.95, 0.90 and 0.80 respectively. Three identical clocks are selected

and cells made by A, B and C are used in clock number 1, 2 and 3 respectively. Let

X be the total number of clocks working after 300 days. Find the probability mass

function (p.m.f.) of X and also plot the distribution function (d.f.) of X.

3. A fair die is rolled independently three times. Define,

Xi ={

1 if the i-th roll yields a perfect square0 otherwise

Find the p.m.f. of Xi. Suppose Y = X1 +X2 +X3. Find the p.m.f. of Y and also it’s

d.f. Find the mean and variance of Y . Verify Chebyshev’s inequality in this case.

4. For what values of k,

fX(x) = (1− k)kx, x = 0, 1, 2, 3, . . .

can serve as a p.m.f. of a random variable X. Find the mean and variance of X.

5. Let

F (x) ={

0 x ≤ 01− 2

3e−

x

3 − 13e−[x

3] x > 0

where [a] means the largest integer ≤ a. Show that F (x) is a d.f. Determine (i)

P (X > 6), (ii) P (X = 5), (iii) P (5 ≤ X ≤ 8).

6. For the d.f.

FX(x) =

0 x < −1x+2

4−1 ≤ x < 1

1 x ≥ 1

sketch the graph FX(x). Obtain the decomposition

FX(x) = c1Fc(x) + c2Fd(x)

where Fc(x) is purely continuous and Fd(x) is purely discrete. Find the mean and

variance of FX(x), Fc(x) and Fd(x).

7. The daily water consumption X (in million of liters) is a random variable with p.d.f.

fX(x) =x

9e−

x

3 , x > 0.

(a) Find the d.f., E(X) and V(X).

(b) Find the probability that on a given day, the water consumption is not more than

6 million liters.

Course: ESO-209

Home Work: 5


1. A mode of a random variable X of the continuous and discrete type is a value that

maximizes the probability density function (p.d.f.) or the probability mass function

(p.m.f.) f(x). If there is only one such x, it is called the mode of the distribution. find

the mode of each of the following distributions:

[a ] f(x) =(

12

)x; x = 1, 2, 3 . . . ..

[b ] f(x) = 12x2(1− x) 0 < x < 1 and zero elsewhere.

[c ] f(x) =(

12

)

x2e−x, 0 < x <∞ and zero elsewhere.

2. A median of a distribution of one random variable X of the discrete or continuous type

is a value x such that P (X ≤ x) < 12and P (X ≤ x) ≥ 1

2. If there is only one such x, it

is called the median of the distribution. Find the median of the following distributions:

[a ] f(x) =(

4x

) (

14

)x (34

)4−x; x = 0, 1, 2, 3, 4. and zero elsewhere.

[b ] f(x) = 3x2, 0 < x < 1 and zero elsewhere.

[c ] f(x) = 1π(1+x2)

, −∞ < x <∞.

3. Let f(x) = 1 for 0 < x < 1 and zero elsewhere, be the p.d.f. ofX. Find the distribution

function and the p.d.f. of Y =√X.

4. Let f(x) = x6, for x = 1,2,3 and zero elsewhere, be the p.m.f. of X. Find the p.m.f. of

Y = X2 and Z = X1+X

.

5. Let f(x) = (4−x)16

; for −2 < x < 2 and zero elsewhere, be the p.d.f. of X. Sketch the

distribution function and p.d.f. of X. If Y = |X|, compute P (Y < 1). If Z = X 2,

compute P (Z < 14).

6. What is the value of∫∞0 xne−xdx, where n is a non-negative integer.

7. Let fX(x) = λe−λx for x > 0 and zero elsewhere. Define Y = [X], the greatest integer

in X. Find the p.m.f. of Y . Evaluate the mean and variance of X and Y .

Course: ESO-209

Home Work: 6


1. If X1 and X2 are random variables of discrete type having the joint probability mass

function

f(x1, x2) =x1 + 2x2

18for(x1, x2) = (1, 1), (1, 2), (2, 1), (2, 2),

and zero elsewhere. Find the marginal p.m.f. of X1 and X2. Also find the conditional

mean and the variance of X2 given X1 = 2.

2. Three balls are placed randomly in 3 boxes B1, B2 and B3. Let N be the total number

boxes which are occupied and X1 be the total number of balls in box Bi. Find the

p.m.f. of (N,X1) and (X1, X2). Find the marginal p.m.f. of X1 in both cases. Can it

be different?

3. Let F (x, y) be a function of two random variables:

F (x, y) ={

0 if y < 0 or x < 0 or x+ y < 11 otherwise

Can this be the distribution function of some random vector (X,Y )? Justify your

answer.

4. Let X1 and X2 have joint p.d.f. f(x1, x2) = x1x2 for 0 < x1 < 1 and 0 < x2 < 1 and

zero elsewhere. Find the conditional mean and the variance of X2 given X1 = x1, for

0 < x1 < 1.

5. If f(x1, x2) = e−(x1+x2) for 0 < x1, x2 < ∞ and zero elsewhere., is the joint p.d.f. of

(X,Y ), show thatX1 and X2 are stochastically independent and also

E(et(X1+X2)) = (1− t)−2, for 0 < t < 1.

Course: ESO-209

Home Work: 7


1. Let X and Y have the joint p.d.f. f(x, y) = 1 for −x < y < x, 0 < x < 1 and zero

elsewhere. Find the graph of E(Y |X = x) as a function of x and also the graph of

E(X|Y = y) as a function of y.

2. Let fX1|X2=x2(x1) = c1x1

x22

for 0 < x1 < x2, 0 < x2 < 1 and zero elsewhere. Also

fX2(x2) = c2x

42 for 0 < x2 < 1 and zero elsewhere. Find the constants c1 and c2.

Determine the joint p.d.f. of X1 and X2. Compute P (0.25 < X1 < 0.5). Also find the

P (0.25 < X1 < 0.5|X2 = 5/8).

3. The random vector (X,Y ) is said to have bivariate normal distribution function, if the

joint p.d.f of (X,Y ) is

fX,Y (x, y) =1

2πσXσY√1− ρ2

×

exp

{

− 1

2(1− ρ2)

[

(

x− µXσX

)2

− 2ρx− µXσX

y − µYσY

+(

y − µYσY

)2]}

for −∞ < x, y <∞, where −1 < ρ < 1, σX , σY > 0.

(a) Show that fX,Y (x, y) is a proper bivariate density function.

(b) Find the joint moment generating function of (X,Y ).

(c) Find the marginal probability density functions of X and Y .

(d) Find the conditional p.d.f of X|Y = y.

(e) Find E(X), E(Y ), V (X), V (Y ), Cov(X,Y ) and Corr(X,Y ).

Course: ESO-209

Home Work: 8


1. For a random vector (X,Y ), show that

E(Y ) = EX(E(Y |X)), V (Y ) = EX(V (Y |X)) + VX(E(Y |X)).

2. Suppose that (X,Y ) has a bivariate normal distribution N2(3, 1, 16, 25, 0.6). Find the

following probabilities: (a) P (3 < Y < 8), (b) P (3 < Y < 8|X = 7), (c) P (−3 < X <

3) and P (−3 < X < 3|Y = 4).

3. Let (X,Y ) has a bivariate norm al distribution N2(20, 10, 1, 25, ρ), where ρ > 0. If

P (4 < Y < 16|X = 20) = 0.954, find ρ.

4. Let X1, . . . X20 be i.i.d. random variables with mean 2 and variance 3. Let

Y =15∑

i=1

Xi Z =20∑

i=11

Xi.

Find E(Y ), V (Y ), E(Z), V (Z) and ρ(Y, Z).

5. Let (X,Y ) be such that E(X) = 15, E(Y ) = 20,, V (X) = 25, V (Y ) = 100, ρ(X,Y )

= -0.6. If U = X − Y and V = 2X − 3Y , find the correlation between U and V .

6. Suppose that the life of light bulbs of certain kind follows the exponential distribution

with mean life 50 hours. Find the probability that among 8 such light bulbs, two

will last less that 4 hours, three will last anywhere from 40 to 60 hours, two will last

anywhere between 60 to 80 hours and one will last more than 80 hours.

Course: ESO-209

Home Work: 9


1 Suppose X1 and X2 are i.i.d uniformly distributed random variables over the intervals

(0, 1). Consider Y1 = X1 +X2 and Y2 = X2 −X1. Find the joint p.d.f. of Y1 and Y2

and also the marginal p.d.f.s of Y1 and Y2.

2 Let X1 and X2 be two independent standard normal random variables. Let Y1 =

X1 + X2 and Y2 = X1/X2. Find the joint p.d.f. of Y1 and Y2 and also the marginal

p.d.f.s of Y1 and Y2.

3. Do the same problem no. 2, assuming that Xi follows gamma(ni, λ).

4. Let X1, X2 and X3 be i.i.d. N(0, 1) random variables. Consider the following random

variables Y1, Y2 and Y3 defined as follows;

X1 = Y1 cosY2 sinY3, X2 = Y1 sinY2 sinY3, X3 = Y1 cosY3

where 0 ≤ Y1 <∞, 0 ≤ Y2 < 2π, 0 ≤ Y3 < π. Find the p.d.f’s of Y1, Y2 and Y3.

5. Let X1, X2 and X3 i.i.d. with p.d.f. f(x) = e−x, 0 < x <∞, zero elsewhere. Find the

p.d.f. of Y1, Y2 and Y3, where

Y1 =X1

X1 +X2

, Y2 =X1 +X2

X1 +X2 +X3

, Y3 = X1 +X2 +X3.

6. Determine the mean and variance of the mean X of a random sample of size 9 from a

distribution having p.d.f. f(x) = 4x3, 0 < x < 1, zero elsewhere.

Course: ESO-209

Home Work: 10


1 Let X1, X2 and X3 be three mutually independent chi-square random variables with

r1, r2 and r3 degrees of freedom, respectively.

(a) Show that Y1 = X1/X2 and Y2 = X1 + X2 are independent and that Y2 is chi-

square random variables with r1 + r2 degrees of freedom.

(b) Find the density functions of

X1/r1X2/r2

andX3/r3

(X1 +X2)/(r1 + r2)

2 Suppose X follows N(0, 1) and Y follows χ2n and they are independent. Find the

density function of Student-t statistic with n degrees of freedom.

T =X√

Yn

.

3. If T had student t distribution with 14 degrees of freedom, find c from the tables such

that P (|T | > c) = 0.05

4. Let f(x) = 1x2 , 1 < x < ∞, zero elsewhere, be the p.d.f. of a random variable X.

consider a random sample of size 72 from the distribution function having this p.d.f.

Compute approximately the probability that more than 50 of the times of the random

sample are less than 3.

5. Let {Xn} be a sequence of i.i.d. random variables from f(x) = e−x, x > 0. Let Yn =

max{X1, . . . , Xn} and Zn = (Yn − lnn). Show that

limn→∞

FZn(z) = FZ(z) = e−e−z

, −∞ < z <∞.

Course: ESO-209

Home Work: 11


1. Let X1, . . . , Xn represent a random sample from each of the distributions having the

following density or mass function:

(1) f(x; θ) = θxe−θ

x!, for x = 0, 1, . . ., 0 < θ < ∞ and it is zero else where. We also

have f(0, 0) = 1.

(2) f(x, θ) = θxθ−1, for 0 < x < 1.

(3) f(x, θ) = e−(x−θ), for θ < x <∞, −∞ < θ <∞ and zero elsewhere.

(4) f(x, θ) = 12e−|x−θ|, for −∞ < x <∞ and zero elsewhere.

2. Let X1, . . . , Xn be a random sample from the following density function

f(x, µ, σ) =1

2√3σ

for µ−√3σ < x < µ+

√3σ

and zero otherwise. Find the maximum likelihood estimators of µ and σ.

3. LetX1, . . . , Xn be a random sample from the distribution function having p.d.f f(x, θ1, θ2) =

1θ2e−x−θ1θ2 , for θ1 ≤ x < ∞, −∞ < θ1 < ∞, 0 < θ2 < ∞ and zero elsewhere. Find the

MLEs of θ1 and θ2.

4. Let X1, . . . , Xn be a random sample from a Gamma(α, λ), find the method of moment

estimators of α and λ and also the MLEs of α and λ.

5. In question no. 1, find the method of moment estimators of the unknown parameters.

Course: ESO-209

Home Work: 12


1. Let Y1, . . . , Yn be the order statistics of a random sample from a distribution function

having p.d.f. f(x, θ) = 1, for θ − 12≤ x ≤ θ + 1

2, −∞ < θ < ∞ and zero elsewhere.

Show that the statistic u(X1, . . . , Xn) such that

Yn −1

2≤ u(X1, . . . , Xn) ≤ Y1 +

1

2

is a maximum likelihood estimator of θ. In particular (4Y1 +2Yn+1)/6 or (Y1 +Yn)/2

are maximum likelihood estimators. Thus uniqueness is not in general a property of a

maximum likelihood estimator.

2. Let X1, X2, X3 have the multinomial distribution in which n = 25, and k = 4, and the

unknown parameters are θ1, θ2 and θ3 respectively i.e. the probability mass function

of X1, X2, X3 is

P (X1 = x1X2 = x2, X3 = x3) =25!

x1!x2!x3!x4!θx1

1 θx2

2 θx3

3 θx4

4 ,

where x4 = 25 − x1 − x2 − x3 and θ4 = 1 − θ1 − θ2 − θ3. If the observed values of

the random variables are x1 = 4, x2 = 11 and x3 = 7, find the maximum likelihood

estimators of θ1, θ2 and θ3

3. LetX1, . . . Xn be i.i.d. N(µ, σ2). Show that the sample average and the sample variance

are unbiased estimators of the population mean and population variance respectively.

4. Let X1 . . . Xn be a random sample from U(0, θ). Find an unbiased of θ based on X(n),

the largest order statistics. Show that X(n) is a consistent estimator of θ.

5. Let Xn ∼ χ2n. Find the limiting distribution of (Xn − n)/

√2n.

hw1

Documents