Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 7 - Probability Distributions [Latest edition]

Chapter 7: Probability Distributions

Exercise 7.1Exercise 7.2Miscellaneous Exercise
Exercise 7.1 [Pages 232 - 233]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Probability Distributions Exercise 7.1 [Pages 232 - 233]

Exercise 7.1 | Q 1 | Page 232

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

Exercise 7.1 | Q 2 | Page 232

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

Exercise 7.1 | Q 3.1 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 P(X) 0.4 0.4 0.2
Exercise 7.1 | Q 3.2 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 3 4 P(X) 0.1 0.5 0.2 − 0.1 0.2
Exercise 7.1 | Q 3.3 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 P(X) 0.1 0.6 0.3
Exercise 7.1 | Q 3.4 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer

 X 3 2 1 0 −1 P(X = x) 0.3 0.2 0.4 0 0.05
Exercise 7.1 | Q 3.5 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 Y −1 0 1 P(Y) 0.6 0.1 0.2
Exercise 7.1 | Q 3.6 | Page 232

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 -1 -2 P(X) 0.3 0.4 0.3
Exercise 7.1 | Q 4.1 | Page 232

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

Exercise 7.1 | Q 4.2 | Page 232

Find the probability distribution of number of tails in the simultaneous tosses of three coins.

Exercise 7.1 | Q 4.3 | Page 232

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

Exercise 7.1 | Q 5 | Page 232

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die

Exercise 7.1 | Q 6 | Page 232

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Exercise 7.1 | Q 7 | Page 232

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Exercise 7.1 | Q 8 | Page 232

A random variable X has the following probability distribution :

 X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine :
(i) k
(ii) P(X < 3)
(iii) P( X > 4)

Exercise 7.1 | Q 9 | Page 232

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

 x -2 -1 0 1 2 P(X) 0.2 0.3 0.1 0.15 0.25
Exercise 7.1 | Q 10 | Page 233

Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.

Exercise 7.1 | Q 11 | Page 233

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

Exercise 7.1 | Q 12 | Page 233

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

Exercise 7.1 | Q 13 | Page 233

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

Exercise 7.1 | Q 14 | Page 233

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

Exercise 7.1 | Q 15 | Page 233

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. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.

Exercise 7.1 | Q 16 | Page 233

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X).

Exercise 7.2 [Pages 238 - 239]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Probability Distributions Exercise 7.2 [Pages 238 - 239]

Exercise 7.2 | Q 1.1 | Page 238

Verify which of the following is p.d.f. of r.v. X:

f(x) = sin x, for 0 ≤ x ≤ π/2

Exercise 7.2 | Q 1.2 | Page 238

Verify which of the following is p.d.f. of r.v. X:

f(x) = x, for 0 ≤ x ≤ 1 and -2 -x for 1 < x < 2

Exercise 7.2 | Q 1.3 | Page 238

Verify which of the following is p.d.f. of r.v. X:

f(x) = 2, for 0 ≤ x ≤ 1.

Exercise 7.2 | Q 2.1 | Page 239

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

f(x) = x/8, for 0 < x < 4 and = 0 otherwise.

Find P (x < 1·5)

Exercise 7.2 | Q 2.2 | Page 239

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

f(x) = x/8, for 0 < x < 4 and = 0 otherwise

P ( 1 < x < 2 )

Exercise 7.2 | Q 2.3 | Page 239

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

f(x) = x/8, for 0 < x < 4 and = 0 otherwise.

P(x > 2)

Exercise 7.2 | Q 3.1 | Page 239

It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = x^2 /3 , for –1 < x < 2 and = 0 otherwise

Verify whether f (x) is p.d.f. of r.v. X.

Exercise 7.2 | Q 3.2 | Page 239

It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = x^2/ 3 , for –1 < x < 2 and = 0 otherwise

Exercise 7.2 | Q 3.3 | Page 239

It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = x^2/3 , for –1 < x < 2 and = 0 otherwise

Find probability that X is negative

Exercise 7.2 | Q 4.1 | Page 239

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

f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P (1/ 4 < x < 3 /2).

Exercise 7.2 | Q 4.2 | Page 239

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

f (x) = kx (1 – x), for 0 < x < 1 and = 0 otherwise, Also find P (1 /4 < x < 1 /2) , P (x < 1 /2).

Exercise 7.2 | Q 5.1 | Page 239

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

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate: P(x≤1)

Exercise 7.2 | Q 5.2 | Page 239

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

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate: P(0.5 ≤ x ≤ 1.5)

Exercise 7.2 | Q 5.3 | Page 239

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

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise. Calculate: P(x ≥ 1.5)

Exercise 7.2 | Q 6.1 | Page 239

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

f (x) = 1/ 5 , for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is between 1 and 3

Exercise 7.2 | Q 6.2 | Page 239

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

f (x) = 1/5 , for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is more than 4 minutes.

Exercise 7.2 | Q 7.1 | Page 239

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

f (x) = k (4 – x^2 ), for –2 ≤ x ≤ 2 and = 0 otherwise.

P(x > 0)

Exercise 7.2 | Q 7.2 | Page 239

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

f (x) = k (4 – x^2 ), for –2 ≤ x ≤ 2 and = 0 otherwise

P(–1 < x < 1)

Exercise 7.2 | Q 7.3 | Page 239

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

f (x) = k (4 – x^2), for –2 ≤ x ≤ 2 and = 0 otherwise.

P (–0·5 < x or x > 0·5)

Exercise 7.2 | Q 8.1 | Page 239

The following is the p.d.f. of continuous r.v.

f (x) = x/8, for 0 < x < 4 and = 0 otherwise.

Find expression for c.d.f. of X

Exercise 7.2 | Q 8.2 | Page 239

The following is the p.d.f. of continuous r.v.

f (x) = x/8 , for 0 < x < 4 and = 0 otherwise.

Find F(x) at x = 0·5 , 1.7 and 5

Exercise 7.2 | Q 9.1 | Page 239

Given the p.d.f. of a continuous r.v. X , f (x) = x^2/3 ,for –1 < x < 2 and = 0 otherwise

Determine c.d.f. of X hence find

P( x < 1)

Exercise 7.2 | Q 9.2 | Page 239

Given the p.d.f. of a continuous r.v. X ,

f (x) = x^2 /3 , for –1 < x < 2 and = 0 otherwise

Determine c.d.f. of X hence find P( x < –2)

Exercise 7.2 | Q 9.3 | Page 239

Given the p.d.f. of a continuous r.v. X ,

f (x) = x^2/ 3 , for –1 < x < 2 and = 0 otherwise

Determine c.d.f. of X hence find P( X > 0)

Exercise 7.2 | Q 9.4 | Page 239

Given the p.d.f. of a continuous r.v. X ,

f (x) = x^2/3 , for –1 < x < 2 and = 0 otherwise

Determine c.d.f. of X hence find P(1 < x < 2)

Exercise 7.2 | Q 10 | Page 239

If a r.v. X has p.d.f.,

f (x) = c /x , for 1 < x < 3, c > 0, Find c, E(X) and Var (X).

Miscellaneous Exercise [Pages 241 - 242]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Probability Distributions Miscellaneous Exercise [Pages 241 - 242]

Miscellaneous Exercise | Q 1 | Page 241

Choose the correct option from the given alternative :

P.d.f. of a.c.r.v X is f (x) = 6x (1 − x), for 0 ≤ x ≤ 1 and = 0, otherwise (elsewhere)

If P (X < a) = P (X > a), then a =

• 1

• 1/2

• 1/3

• 1/4

Miscellaneous Exercise | Q 2 | Page 242

Choose the correct option from the given alternative:

If the p.d.f of a.c.r.v. X is f (x) = 3 (1 − 2x2 ), for 0 < x < 1 and = 0, otherwise (elsewhere) then the c.d.f of X is F(x) =

• 2x − 3x2

• 3x − 4x3

• 3x − 2x3

• 2x3 − 3x

Miscellaneous Exercise | Q 3 | Page 242

Choose the correct option from the given alternative:

If the p.d.f of a.c.r.v. X is f (x) = x^2/ 18 , for −3 < x < 3 and = 0, otherwise then P (| X | < 1) =

• 1/27

• 1/28

• 1/29

• 1/26

Miscellaneous Exercise | Q 4 | Page 242

Choose the correct option from the given alternative:

If a d.r.v. X takes values 0, 1, 2, 3, . . . which probability P (X = x) = k (x + 1)·5 −x , where k is a constant, then P (X = 0) =

• 7/25

• 16/25

• 18/25

• 19/25

Miscellaneous Exercise | Q 5 | Page 242

Choose the correct option from the given alternative:

If p.m.f. of a d.r.v. X is P (X = x) = ((c_(x)^5 ))/2^5 , for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise If a = P (X ≤ 2) and b = P (X ≥ 3), then E (X ) =

• a < b

• a > b

• a = b

• a + b

Miscellaneous Exercise | Q 6 | Page 242

Choose the correct option from the given alternative:

If p.m.f. of a d.r.v. X is P (X = x) = x^2 /(n (n + 1)), for x = 1, 2, 3, . . ., n and = 0, otherwise then E (X ) =

• n/ 1 + 1/ 2

• n /3 + 1 /6

• n/ 2 + 1 /5

• n /1 + 1/ 3

Miscellaneous Exercise | Q 7 | Page 242

Choose the correct option from the given alternative :

If p.m.f. of a d.r.v. X is P (x) = c/ x^3 , for x = 1, 2, 3 and = 0, otherwise (elsewhere) then E (X ) =

• 343/ 297

• 294 /251

• 297 /294

• 294 /297

Miscellaneous Exercise | Q 8 | Page 242

Choose the correct option from the given alternative:

If the a d.r.v. X has the following probability distribution :

 x -2 -1 0 1 2 3 p(X=x) 0.1 k 0.2 2k 0.3 k

then P (X = −1) =

• 1/10

• 2/10

• 3/10

• 4/10

Miscellaneous Exercise | Q 9 | Page 242

Choose the correct option from the given alternative:

If the a d.r.v. X has the following probability distribution:

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

k =

• 1/7

• 1/8

• 1/9

• 1/10

Miscellaneous Exercise | Q 10 | Page 242

Choose the correct option from the given alternative:

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

 X -2 -1 0 1 2 P(x) 0.3 0.4 0.2 0.15 0.25
• 0·85

• – 0·35

• 0·15

• – 0·15

Miscellaneous Exercise [Pages 242 - 244]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Probability Distributions Miscellaneous Exercise [Pages 242 - 244]

Miscellaneous Exercise | Q 1.1 | Page 242

Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

An economist is interested the number of unemployed graduate in the town of population 1 lakh.

Miscellaneous Exercise | Q 1.2 | Page 242

Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

Amount of syrup prescribed by physician.

Miscellaneous Exercise | Q 1.3 | Page 242

Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

The person on the high protein diet is interested gain of weight in a week.

Miscellaneous Exercise | Q 1.4 | Page 242

Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day

Miscellaneous Exercise | Q 1.5 | Page 242

Solve the following:

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

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

Miscellaneous Exercise | Q 2 | Page 242

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

 x = x 1 2 3 4 5 6 P[x=x] k 2k 3k 4k 5k 6k

(i) Determine the value of k.

(ii) Find P(X≤4), P(2<X< 4), P(X≥3).

Miscellaneous Exercise | Q 3.1 | Page 242

Solve the following :

The following probability distribution of r.v. X

 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

Miscellaneous Exercise | Q 3.2 | Page 242

Solve the following :

The following probability distribution of r.v. X

 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

Miscellaneous Exercise | Q 3.3 | Page 242

Solve the following :

The following probability distribution of r.v. X

 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

Miscellaneous Exercise | Q 3.4 | Page 242

Solve the following :

The following probability distribution of r.v. X

 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

Miscellaneous Exercise | Q 4 | Page 242

The p.m.f. of a r.v. X is given by P (X = x) =("" ^5 C_x ) /2^5 , for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise.

Then show that P (X ≤ 2) = P (X ≥ 3).

Miscellaneous Exercise | Q 5 | Page 242

In the p.m.f. of r.v. X

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

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

Miscellaneous Exercise | Q 6 | Page 242

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.

Miscellaneous Exercise | Q 7 | Page 244

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as

(i) number greater than 4

(ii) six appears on at least one die

Miscellaneous Exercise | Q 8 | Page 244

A random variable X has the following probability distribution :

 x = x 0 1 2 3 7 P(X=x) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine (i) k

(ii) P(X> 6)

(iii) P(0<X<3).

Miscellaneous Exercise | Q 9.1 | Page 244

The following is the c.d.f. of 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 p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).

Miscellaneous Exercise | Q 9.2 | Page 244

The following is the c.d.f. of 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

P (–1 ≤ X ≤ 2)

Miscellaneous Exercise | Q 9.3 | Page 244

The following is the c.d.f. of 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

P (X ≤ 3/ X > 0)

Miscellaneous Exercise | Q 10.1 | Page 244

Find the expected value, variance and standard deviation of the random variable whose p.m.f.’s are given below :

 x = x 1 2 3 P (X = x) 1/5 2/5 2/5
Miscellaneous Exercise | Q 10.2 | Page 244

Find the expected value, variance and standard deviation of the random variable whose p.m.f.’s are given below :

 x = x -1 0 3 P (X = x) 1/5 2/5 2/5
Miscellaneous Exercise | Q 10.3 | Page 244

Find the expected value, variance and standard deviation of the random variable whose p.m.f.’s are given below :

 x = x 1 2 3 ... n P (X = x) 1/n 1/n 1/n ... 1/n
Miscellaneous Exercise | Q 10.4 | Page 244

Find the expected value, variance, and standard deviation of the random variable whose p.m.f.’s are given below :

 x = x 0 1 2 3 4 5 P (X = x) 1/32 5/32 10/32 10/32 5/32 1/32
Miscellaneous Exercise | Q 11 | Page 244

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.

Miscellaneous Exercise | Q 12 | Page 244

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

P (x) = (3 – x) /1 = 0 , for x = –1, 0, 1, 2 and = 0, otherwise

Calculate E(X ) and Var (X ).

Miscellaneous Exercise | Q 13.1 | Page 244

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

f (x) = k (4 – x^2 ), for –2 ≤ x ≤ 2 and = 0 otherwise.

P(x > 0)

Miscellaneous Exercise | Q 13.2 | Page 244

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

f (x) = k (4 – x^2 ), for –2 ≤ x ≤ 2 and = 0 otherwise

P(–1 < x < 1)

Miscellaneous Exercise | Q 13.3 | Page 244

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

f (x) = k (4 – x^2), for –2 ≤ x ≤ 2 and = 0 otherwise.

P (–0·5 < x or x > 0·5)

Miscellaneous Exercise | Q 14 | Page 244

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

f (x) = 1/ (2a) , for 0 < x < 2a and = 0, otherwise. Show that P [X < a/ 2] = P [X >( 3a)/ 2] .

Miscellaneous Exercise | Q 15 | Page 244

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

f (x) = k /sqrtx , for 0 < x < 4 and = 0, otherwise. Determine k .

Determine c.d.f. of X and hence P (X ≤ 2) and P(X ≤ 1).

Chapter 7: Probability Distributions

Exercise 7.1Exercise 7.2Miscellaneous Exercise

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 7 - Probability Distributions

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 7 Probability Distributions are Random Variables and Its Probability Distributions, Types of Random Variables, Probability Distribution of Discrete Random Variables, Probability Distribution of a Continuous Random Variable, Variance of a Random Variable.

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