# Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 - Probability Distributions [Latest edition]

## Chapter 8: Probability Distributions

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4Miscellaneous Exercise 8Part IPart II
Exercise 8.1 [Pages 140 - 141]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Exercise 8.1 [Pages 140 - 141]

Exercise 8.1 | Q 1.01 | Page 140

Let X represent the difference between number of heads and number of tails obtained when a coin is tossed 6 times. What are the possible values of X?

Exercise 8.1 | Q 1.02 | Page 140

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are the possible values of X?

Exercise 8.1 | Q 1.03 | Page 140

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 x 0 1 2 P(x) 0.4 0.4 0.2
Exercise 8.1 | Q 1.03 | Page 140

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 x 0 1 2 3 4 P(x) 0.1 0.5 0.2 –0.1 0.3
Exercise 8.1 | Q 1.03 | Page 140

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 x 0 1 2 P(x) 0.1 0.6 0.3
Exercise 8.1 | Q 1.03 | Page 140

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 z 3 2 1 0 -1 P(z) 0.3 0.2 0.4. 0.05 0.05
Exercise 8.1 | Q 1.03 | Page 141

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 y –1 0 1 P(y) 0.6 0.1 0.2
Exercise 8.1 | Q 1.03 | Page 141

Determine whether each of the following is a probability distribution. Give reasons for your answer.

 x 0 1 2 P(x) 0.3 0.4 0.2
Exercise 8.1 | Q 1.04 | Page 141

Find the probability distribution of number of heads in two tosses of a coin,

Exercise 8.1 | Q 1.04 | Page 141

Find the probability distribution of number of number of tails in three tosses of a coin

Exercise 8.1 | Q 1.04 | Page 141

Find the probability distribution of number of heads in four tosses of a coin

Exercise 8.1 | Q 1.05 | Page 141

Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.

Exercise 8.1 | Q 1.06 | Page 141

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.

Exercise 8.1 | Q 1.07 | Page 141

A coin is biased so that the head is 3 times as likely to occur as tail. Find the probability distribution of number of tails in two tosses.

Exercise 8.1 | Q 1.08 | Page 141

A random variable X has the following probability distribution:

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine k

Exercise 8.1 | Q 1.08 | Page 141

A random variable X has the following probability distribution:

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(X < 3)

Exercise 8.1 | Q 1.08 | Page 141

A random variable X has the following probability distribution:

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(0 < X < 3)

Exercise 8.1 | Q 1.08 | Page 141

A random variable X has the following probability distribution:

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(X > 4)

Exercise 8.1 | Q 1.09 | Page 141

Find expected value and variance of X using the following p.m.f.

 X –2 –1 0 1 2 P(x) 0.2 0.3 0.1 0.15 0.25
Exercise 8.1 | Q 1.1 | Page 141

Find expected value and variance of X, the number on the uppermost face of a fair die.

Exercise 8.1 | Q 1.11 | Page 141

Find the mean of number of heads in three tosses of a fair coin

Exercise 8.1 | Q 1.12 | Page 141

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X

Exercise 8.1 | Q 1.13 | Page 141

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

Exercise 8.1 | Q 1.14 | Page 141

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of X.

Exercise 8.1 | Q 1.15 | Page 141

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.

Exercise 8.1 | Q 1.16 | Page 141

70% of the members favour and 30% oppose a proposal in a meeting. The random variable X takes the value 0 if a member opposes the proposal and the value 1 if a member is in favour. Find E(X) and Var(X).

Exercise 8.2 [Pages 144 - 145]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Exercise 8.2 [Pages 144 - 145]

Exercise 8.2 | Q 1.01 | Page 144

Check whether the following is a p.d.f.

f(x) = {(x, "for"  0 ≤ x ≤ 1),(2 - x, "for"  1 < x ≤ 2.):}

Exercise 8.2 | Q 1.01 | Page 144

Check whether the following is a p.d.f.

f(x) = 2  for 0 < x < q.

Exercise 8.2 | Q 1.02 | Page 144

The following is the p.d.f. of a r.v. X.

f(x) = {(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}

Find P(X < 1.5),

Exercise 8.2 | Q 1.02 | Page 144

The following is the p.d.f. of a r.v. X.

f(x) = {(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}

Find P(1 < X < 2),

Exercise 8.2 | Q 1.02 | Page 144

The following is the p.d.f. of a r.v. X.

f(x) = {(x/(8),  "for"  0 < x < 4),(0,  "otherwise."):}

Find P(X > 2)

Exercise 8.2 | Q 1.03 | Page 144

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r. v. with p. d. f.

f(x) = {(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}
Verify whether f(x) is a p.d.f.

Exercise 8.2 | Q 1.03 | Page 144

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r. v. with p. d. f.

f(x) = {(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}
Find P(0 < X ≤ 1).

Exercise 8.2 | Q 1.03 | Page 144

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r. v. with p. d. f.

f(x) = {(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}
Find probability that X is between 1 and 3..

Exercise 8.2 | Q 1.04 | Page 144

Find k if the following function represents the p. d. f. of a r. v. X.

f(x) = {(kx,  "for"  0 < x < 2),(0,  "otherwise."):}

Also find "P"[1/4 < "X" < 1/2]

Exercise 8.2 | Q 1.04 | Page 145

Find k if the following function represents the p. d. f. of a r. v. X.

f(x) = {(kx(1 - x), "for"  0 < x < 1),(0,  "otherwise".):}
Also Find
a. "P"[1/4  "X" < 1/2]

b. "P"["X" < 1/2]

Exercise 8.2 | Q 1.05 | Page 145

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = {(0.5x,  "for" 0 ≤ x ≤ 2),(0,  "otherwise".):}
Calculate : P(X ≤ 1)

Exercise 8.2 | Q 1.05 | Page 145

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = {(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}
Calculate : P(0.5 ≤ X ≤ 1.5)

Exercise 8.2 | Q 1.05 | Page 145

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = {(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}
Calculate : P(X ≥ 1.5)

Exercise 8.2 | Q 1.06 | Page 145

Suppose X is the waiting time (in minutes) for a bus and its p. d. f. is given by

f(x) = {(1/5,  "for"  0 ≤ x ≤ 5),(0,  "otherwise".):}
Find the probability that waiting time is between 1 and 3 minutes

Exercise 8.2 | Q 1.06 | Page 145

Suppose X is the waiting time (in minutes) for a bus and its p. d. f. is given by

f(x) = {(1/5,  "for"  0 ≤ x ≤ 5),(0,  "otherwise".):}
Find the probability that waiting time is more than 4 minutes.

Exercise 8.2 | Q 1.07 | Page 145

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = {("k"(4 - x^2),  "for" -2 ≤ x ≤ 2),(0,  "otherwise".):}
compute P(X > 0)

Exercise 8.2 | Q 1.07 | Page 145

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = {("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}
compute P(–1 < X < 1)

Exercise 8.2 | Q 1.07 | Page 145

Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.

f(x) = {("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}
compute P(X < – 0.5 or X > 0.5)

Exercise 8.2 | Q 1.08 | Page 145

Following is the p. d. f. of a continuous r.v. X.

f(x) = {(x/8,  "for"  0 < x < 4),(0,  "otherwise".):}
Find expression for the c.d.f. of X.

Exercise 8.2 | Q 1.08 | Page 145

Following is the p. d. f. of a continuous r.v. X.

f(x) = {(x/8,  "for"  0 < x < 4),(0,  "otherwise".):}
Find F(x) at x = 0.5, 1.7 and 5.

Exercise 8.2 | Q 1.09 | Page 145

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8),  0 < x < 2),(0,   "otherwise".):}
Determine the c.d.f. of X and hence find P(X < 1)

Exercise 8.2 | Q 1.09 | Page 145

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(X < –2)

Exercise 8.2 | Q 1.09 | Page 145

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8),  0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(X > 0)

Exercise 8.2 | Q 1.09 | Page 145

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(1 < X < 2)

Exercise 8.2 | Q 1.1 | Page 145

If a r.v. X has p.d.f f(x) = {("c"/x","  1 < x < 3"," "c" > 0),(0","  "otherwise"):}
Find c, E(X), and Var(X). Also Find F(x).

Exercise 8.3 [Pages 150 - 151]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Exercise 8.3 [Pages 150 - 151]

Exercise 8.3 | Q 1.01 | Page 150

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes

Exercise 8.3 | Q 1.01 | Page 150

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes

Exercise 8.3 | Q 1.01 | Page 150

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes

Exercise 8.3 | Q 1.02 | Page 150

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes

Exercise 8.3 | Q 1.03 | Page 150

There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

Exercise 8.3 | Q 1.04 | Page 150

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that all the five cards are spades.

Exercise 8.3 | Q 1.04 | Page 150

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that only 3 cards are spades

Exercise 8.3 | Q 1.04 | Page 150

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that none is a spade.

Exercise 8.3 | Q 1.05 | Page 150

The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

Exercise 8.3 | Q 1.06 | Page 150

10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?

Exercise 8.3 | Q 1.07 | Page 151

In a multiple choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?

Exercise 8.3 | Q 1.08 | Page 151

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Exercise 8.3 | Q 1.09 | Page 151

Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)

Exercise 8.3 | Q 1.09 | Page 151

Given that X ~ B(n,p), if p = 0.6 and E(X) = 6, find n and Var(X).

Exercise 8.3 | Q 1.09 | Page 151

Given that X ~ B(n,p), if n = 25, E(X) = 10, find p and Var (X).

Exercise 8.3 | Q 1.09 | Page 151

Given that X ~ B(n,p), if n = 10, E(X) = 8, find Var(X).

Exercise 8.4 [Page 152]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Exercise 8.4 [Page 152]

Exercise 8.4 | Q 1.01 | Page 152

If X has Poisson distribution with m = 1, then find P(X ≤ 1) given e−1 = 0.3678

Exercise 8.4 | Q 1.02 | Page 152

If X~P(0.5), then find P(X = 3) given e−0.5 = 0.6065.

Exercise 8.4 | Q 1.03 | Page 152

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e−3 = 0.0497

Exercise 8.4 | Q 1.04 | Page 152

The number of complaints which a bank manager receives per day follows a Poisson distribution with parameter m = 4. Find the probability that the manager receives a) only two complaints on a given day, b) at most two complaints on a given day. Use e−4 = 0.0183.

Exercise 8.4 | Q 1.05 | Page 152

A car firm has 2 cars, which are hired out day by day. The number of cars hired on a day follows Poisson distribution with mean 1.5. Find the probability that (i) no car is used on a given day, (ii) some demand is refused on a given day, given e−1.5 = 0.2231.

Exercise 8.4 | Q 1.06 | Page 152

Defects on plywood sheet occur at random with the average of one defect per 50 sq. ft. Find the probability that such a sheet has (i) no defect, (ii) at least one defect. Use e−1 = 0.3678.

Exercise 8.4 | Q 1.07 | Page 152

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has exactly 5 rats inclusive. Given e-5  = 0.0067.

Exercise 8.4 | Q 1.07 | Page 152

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has more than 5 rats inclusive. Given e-5  = 0.0067.

Exercise 8.4 | Q 1.07 | Page 152

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has between 5 and 7 rats inclusive. Given e−5 = 0.0067.

Miscellaneous Exercise 8 [Pages 153 - 154]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Miscellaneous Exercise 8 [Pages 153 - 154]

Miscellaneous Exercise 8 | Q 1.01 | Page 153

Choose the correct alternative :

F(x) is c.d.f. of discrete r.v. X whose p.m.f. is given by P(x) = "k"[(4),(x)] , for x = 0, 1, 2, 3, 4 and P(x) = 0 otherwise then F(5) = _______

• (1)/(16)

• (1)/(8)

• (1)/(4)

• 1

Miscellaneous Exercise 8 | Q 1.02 | Page 153

Choose the correct alternative :

F(x) is c.d.f. of discrete r.v. X whose distribution is

 Xi – 2 – 1 0 1 2 Pi 0.2 0.3 0.15 0.25 0.1

Then F(–  3) = _______ .

• 0

• 1

• 0.2

• 0.15

Miscellaneous Exercise 8 | Q 1.03 | Page 153

Choose the correct alternative :

X: is number obtained on upper most face when a fair die….thrown then E(X) = _______.

• 3.0

• 3.5

• 4.0

• 4.5

Miscellaneous Exercise 8 | Q 1.04 | Page 153

Choose the correct alternative :

If p.m.f. of r.v.X is given below.

 x 0 1 2 P(x) q2 2pq p2

Then Var(X) = _______

• p2

• q

• pq

• 2pq

Miscellaneous Exercise 8 | Q 1.05 | Page 153

Choose the correct alternative :

The expected value of the sum of two numbers obtained when two fair dice are rolled is _______.

• 5

• 6

• 7

• 8

Miscellaneous Exercise 8 | Q 1.06 | Page 153

Choose the correct alternative :

Given p.d.f. of a continuous r.v.X as f(x) =  x^2/(3) for –1 < x < 2 = 0 otherwise then F(1) = _______.

• (1)/(9)

• (2)/(9)

• (3)/(9)

• (4)/(9)

Miscellaneous Exercise 8 | Q 1.07 | Page 153

Choose the correct alternative :

X is r.v. with p.d.f. f(x) = "k"/sqrt(x), 0 < x < 4 = 0 otherwise then x E(X) = _______

• (1)/(3)

• (4)/(3)

• (2)/(3)

• 1

Miscellaneous Exercise 8 | Q 1.08 | Page 153

Choose the correct alternative :

If X ∼ B(20, 1/10) then E(X) = _______

• 2

• 5

• 4

• 3

Miscellaneous Exercise 8 | Q 1.09 | Page 153

Choose the correct alternative :

If E(X) = m and Var (X) = m then X follows ______ .

• Binomial distribution

• Poisson distribution

• Normal distribution

• Normal distribution

Miscellaneous Exercise 8 | Q 1.1 | Page 154

Choose the correct alternative :

If E(X) > Var (X) then X follows _______ .

• Binomial distribution

• Poisson distribution

• Normal distribution

• None of the above

Miscellaneous Exercise 8 | Q 2.01 | Page 154

Fill in the blank :

The values of discrete r.v. are generally obtained by _______

Miscellaneous Exercise 8 | Q 2.02 | Page 154

Fill in the blank :

The value of continuous r.v. are generally obtained by _______

Miscellaneous Exercise 8 | Q 2.03 | Page 154

Fill in the blank :

If X is discrete random variable takes the value x1, x2, x3,…, xn then $\sum\limits_{i=1}^{n}\text{P}(x_i)$ = _______

Miscellaneous Exercise 8 | Q 2.04 | Page 154

Fill in the blank :

If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = (x - 1)/(3) for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______

Miscellaneous Exercise 8 | Q 2.05 | Page 154

Fill in the blank :

If F(x) is distribution function of discrete r.v.X with p.m.f. P(x) = "k"[(4),(x)] for x = 0, 1, 2, 3, 4 and P(x) = 0 otherwise then F(–1) = _______

Miscellaneous Exercise 8 | Q 2.06 | Page 154

Fill in the blank :

E(x) is considered to be _______ of the probability distribution of x.

Miscellaneous Exercise 8 | Q 2.07 | Page 154

Fill in the blank :

If x is continuous r.v. and F(xi) = P(X ≤ xi) = int_(-oo)^(oo) f(x)*dx then F(x) is called _______

Miscellaneous Exercise 8 | Q 2.08 | Page 154

Fill in the blank :

In Binomial distribution probability of success Remains constant / independent from trial to trial.

Miscellaneous Exercise 8 | Q 2.09 | Page 154

Fill in the blank :

In Binomial distribution if n is very large and probability success of p is very small such that np = m (constant) then _______ distribution is applied.

Miscellaneous Exercise 8 | Q 3.01 | Page 154

State whether the following is True or False :

If P(X = x) = "k"[(4),(x)] for x = 0, 1, 2, 3, 4 , then F(5) = (1)/(4) when F(x) is c.d.f.

• True

• False

Miscellaneous Exercise 8 | Q 3.02 | Page 154

State whether the following is True or False :

 x – 2 – 1 0 1 2 P(X = x) 0.2 0.3 0.15 0.25 0.1

If F(x) is c.d.f. of discrete r.v. X then F(–3) = 0

• True

• False

Miscellaneous Exercise 8 | Q 3.03 | Page 154

State whether the following is True or False :

X is the number obtained on upper most face when a die is thrown then E(X) = 3.5.

• True

• False

Miscellaneous Exercise 8 | Q 3.04 | Page 154

State whether the following is True or False :

If p.m.f. of discrete r.v. X is

 x 0 1 2 P(X = x) q2 2pq p2

then E(x) = 2p.

• True

• False

Miscellaneous Exercise 8 | Q 3.05 | Page 154

State whether the following is True or False :

The p.m.f. of a r.v. X is P(x) = (2x)/("n"("n" + 1)) , x = 1, 2, ……. n
= 0                  ,otherwise
Then E(x) = (2"n" + 1)/(3)

• True

• False

Miscellaneous Exercise 8 | Q 3.06 | Page 154

State whether the following is True or False :

If f(x) = k x (1 – x) for 0 < x < 1 = 0 otherwise k = 12

• True

• False

Miscellaneous Exercise 8 | Q 3.07 | Page 154

State whether the following is True or False :

If X ~ B(n,p) and n = 6 and P(X = 4) = P(X = 2) then p = (1)/(2)

• True

• False

Miscellaneous Exercise 8 | Q 3.08 | Page 154

State whether the following is True or False :

If r.v. X assumes values 1, 2, 3, ……. n with equal probabilities then E(X) = ("n" + 1)/(2)

• True

• False

Miscellaneous Exercise 8 | Q 3.09 | Page 154

State whether the following is True or False :

If r.v. X assumes the values 1, 2, 3, ……. 9 with equal probabilities, E(x) = 5.

• True

• False

Part I [Pages 155 - 156]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Part I [Pages 155 - 156]

Part I | Q 1.01 | Page 155

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

An economist is interested in knowing the number of unemployed graduates in the town with a population of 1 lakh.

Part I | Q 1.01 | Page 155

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

Amount of syrup prescribed by a physician.

Part I | Q 1.01 | Page 155

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

A person on high protein diet is interested in the weight gained in a week.

Part I | Q 1.01 | Page 155

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

Twelve of 20 white rats available for an experiment are male. A scientist randomly selects 5 rats and counts the number of female rats among them.

Part I | Q 1.01 | Page 155

Solve the following problem :

Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.

A highway safety group is interested in the speed (km/hrs) of a car at a check point.

Part I | Q 1.02 | Page 155

Solve the following problem :

The probability distribution of a discrete r.v. X is as follows.

 X 1 2 3 4 5 6 (X = x) k 2k 3k 4k 5k 6k

Determine the value of k.

Part I | Q 1.02 | Page 155

Solve the following problem :

The probability distribution of a discrete r.v. X is as follows.

 X 1 2 3 4 5 6 (X = x) k 2k 3k 4k 5k 6k

Find P(X ≤ 4), P(2 < X < 4), P(X ≤ 3).

Part I | Q 1.03 | Page 155

Solve the following problem :

Following is the probability distribution of a r.v.X.

 X – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is positive.

Part I | Q 1.03 | Page 155

Solve the following problem :

Following is the probability distribution of a r.v.X.

 x – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is non-negative

Part I | Q 1.03 | Page 155

Solve the following problem:

Following is the probability distribution of a r.v.X.

 X – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is odd.

Part I | Q 1.03 | Page 155

Solve the following problem :

Following is the probability distribution of a r.v.X.

 x – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is even.

Part I | Q 1.04 | Page 155

Solve the following problem :

The p.m.f. of a r.v.X is given by

P(X = x) = {(((5),(x)) 1/2^5", ", x = 0", "1", "2", "3", "4", "5.),(0,"otherwise"):}

Show that P(X ≤ 2) = P(X ≤ 3).

Part I | Q 1.05 | Page 155

Solve the following problem :

In the following probability distribution of a r.v.X.

 x 1 2 3 4 5 P (x) (1)/(20) (3)/(20) a 2a (1)/(20)

Find a and obtain the c.d.f. of X.

Part I | Q 1.06 | Page 155

Solve the following problem :

A fair coin is tossed 4 times. Let X denote the number of heads obtained. Identify the probability distribution of X and state the formula for p. m. f. of X.

Part I | Q 1.07 | Page 155

Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as number greater than 4.

Part I | Q 1.07 | Page 155

Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.

Part I | Q 1.08 | Page 155

Solve the following problem :

A random variable X has the following probability distribution.

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine k

Part I | Q 1.08 | Page 155

Solve the following problem :

A random variable X has the following probability distribution.

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(X < 3)

Part I | Q 1.08 | Page 155

Solve the following problem :

A random variable X has the following probability distribution.

 x 1 2 3 4 5 6 7 F (x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(X > 6)

Part I | Q 1.08 | Page 155

Solve the following problem:

A random variable X has the following probability distribution.

 x 1 2 3 4 5 6 7 P(x) k 2k 2k 3k k2 2k2 7k2 + k

Determine P(0 < X < 3)

Part I | Q 1.09 | Page 155

Solve the following problem :

The following is the c.d.f of a r.v.X.

 x – 3 – 2 – 1 0 1 2 3 4 F (x) 0.1 0.3 0.5 0.65 0.75 0.85 0.9 1

Find the probability distribution of X and P(–1 ≤ X ≤ 2).

Part I | Q 1.1 | Page 155

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

 x 1 2 3 P(X = x) (1)/(5) (2)/(5) (2)/(5)
Part I | Q 1.1 | Page 156

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

 x – 1 0 1 P(X = x) (1)/(5) (2)/(5) (2)/(5)
Part I | Q 1.1 | Page 156

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

 x 1 2 3 ... n P(X = x) (1)/"n" (1)/"n" (1)/"n" ... (1)/"n"
Part I | Q 1.1 | Page 156

Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

 X 0 1 2 3 4 5 P(X = x) (1)/(32) (5)/(32) (10)/(32) (10)/(32) (5)/(32) (1)/(32)
Part I | Q 1.11 | Page 156

Solve the following problem :

A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.

Part I | Q 1.12 | Page 156

Solve the following problem :

Let the p. m. f. of the r. v. X be

"P"(x) = {((3 - x)/(10)", ","for"  x = -1", "0", "1", "2.),(0,"otherwise".):}
Calculate E(X) and Var(X).

Part I | Q 1.13 | Page 156

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = {("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}
Compute P(X > 0)

Part I | Q 1.13 | Page 156

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = {("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}
Compute P(–1 < X < 1)

Part I | Q 1.13 | Page 156

Solve the following problem :

Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.

f(x) = {("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}
Compute P(X < – 0.5 or X > 0.5)

Part I | Q 1.14 | Page 156

Solve the following problem :

The p.d.f. of the r.v. X is given by

f(x) = {((1)/(2"a")",", "for"  0 <  x= 2"a".),(0, "otherwise".):}
Show that "P"("X" < "a"/2) = "P"("X" > (3"a")/2)

Part I | Q 1.15 | Page 156

Solve the following problem :

Determine k if the p.d.f. of the r.v. is

f(x) = {("ke"^(-thetax),  "for"  0 ≤ x < oo),(0, "otherwise".):}
Find "P"("X" > 1/theta) and determine M is P(0 < X < M) = (1)/(2)

Part I | Q 1.16 | Page 156

Solve the following problem :

The p.d.f. of the r.v. X is given by

f(x) = {("k"/sqrt(x), "for"  0 < x < 4.),(0, "otherwise".):}
Determine k, the c.d.f. of X, and hence find P(X ≤ 2) and P(X ≥ 1).

Part I | Q 1.17 | Page 156

Solve the following problem :

Let X denote the reaction temperature in Celsius of a certain chemical process. Let X have the p. d. f.

f(x) = {((1)/(10),  "for" -5 ≤ x < 5),(0, "otherwise".):}
Compute P(X < 0).

Part II [Pages 156 - 157]

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Probability Distributions Part II [Pages 156 - 157]

Part II | Q 1.01 | Page 156

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X = 1)

Part II | Q 1.01 | Page 156

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X ≥1)

Part II | Q 1.01 | Page 156

Solve the following problem :

Let X ∼ B(10,0.2). Find P(X ≤ 8).

Part II | Q 1.02 | Page 156

Solve the following problem :

Let X∼B(n,p) If n = 10 and E(X)= 5, find p and Var(X).

Part II | Q 1.02 | Page 156

Solve the following problem :

Let X∼B(n,p) If E(X) = 5 and Var(X) = 2.5, find n and p.

Part II | Q 1.03 | Page 156

Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows 3 heads

Part II | Q 1.03 | Page 156

Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows head in the first 2 tosses and tail in last 2 tosses.

Part II | Q 1.04 | Page 156

Solve the following problem :

The probability that a bomb will hit the target is 0.8. Find the probability that, out of 5 bombs, exactly 2 will miss the target.

Part II | Q 1.05 | Page 156

Solve the following problem :

The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.

Part II | Q 1.06 | Page 156

Solve the following problem :

A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Find the probability that the inspector finds at most one defective item in the 4 selected items.

Part II | Q 1.07 | Page 157

Solve the following problem :

The probability that a component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 components tested survive.

Part II | Q 1.08 | Page 157

Solve the following problem :

An examination consists of 5 multiple choice questions, in each of which the candidate has to decide which one of 4 suggested answers is correct. A completely unprepared student guesses each answer completely randomly. Find the probability that,

1. the student gets 4 or more correct answers.
2. the student gets less than 4 correct answers.
Part II | Q 1.09 | Page 157

Solve the following problem :

The probability that a machine will produce all bolts in a production run within the specification is 0.9. A sample of 3 machines is taken at random. Calculate the probability that all machines will produce all bolts in a production run within the specification.

Part II | Q 1.1 | Page 157

Solve the following problem :

A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 0

Part II | Q 1.1 | Page 157

Solve the following problem :

A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 1 terminal requires attention during a week.

Part II | Q 1.11 | Page 157

Solve the following problem :

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.

Calculate the probabilities of obtaining an answer yes from all of the selected students.

Part II | Q 1.11 | Page 157

Solve the following problem :

In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.

Find the probability that the visitor obtains the answer yes from at least 3 students.

Part II | Q 1.12 | Page 157

Solve the following problem :

It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.

Part II | Q 1.12 | Page 157

Solve the following problem :

It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.

Part II | Q 1.13 | Page 157

Solve the following problem :

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, find variance of X.

Part II | Q 1.14 | Page 157

Solve the following problem :

If X follows Poisson distribution with parameter m such that
("P"("X" = x + 1))/("P"("X" = x)) = (2)/(x + 1)
Find mean and variance of X.

## Chapter 8: Probability Distributions

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4Miscellaneous Exercise 8Part IPart II

## Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 - Probability Distributions

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Concepts covered in Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 Probability Distributions are Mean of a Random Variable, Types of Random Variables, Random Variables and Its Probability Distributions, Probability Distribution of Discrete Random Variables, Probability Distribution of a Continuous Random Variable, Binomial Distribution, Bernoulli Trial, Mean of Binomial Distribution (P.M.F.), Variance of Binomial Distribution (P.M.F.), Poisson Distribution.

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