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## Solutions for Chapter 7: Probability Distributions

Below listed, you can find solutions for Chapter 7 of Maharashtra State Board Balbharati for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board.

### 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]

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?

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?

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 |

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 |

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 |

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(Z) |
0.3 | 0.2 | 0.4 | 0 | 0.05 |

Y |
−1 | 0 | 1 |

P(Y) |
0.6 | 0.1 | 0.2 |

X | 0 | -1 | -2 |

P(X) | 0.3 | 0.4 | 0.3 |

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

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

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

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.

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.

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.

**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 | k^{2} |
2k^{2} |
7k^{2} + k |

**Determine:**

- k
- P(X < 3)
- P( X > 4)

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 |

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

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

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

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).

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

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.

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).

### 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]

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

f(x) = sin x, for 0 ≤ x ≤ `π/2`

**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

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

f(x) = 2, for 0 ≤ x ≤ 1.

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)

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 )

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)

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.

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

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

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)`.

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)`.

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)

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)

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)

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

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.

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)

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,"),(0 "otherwise".):}`

P(–1 < x < 1)

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)

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

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

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)

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)

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)

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)

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).

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 7 Probability Distributions Miscellaneous Exercise 1 [Pages 241 - 242]

**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`

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 − 3x

^{2}3x − 4x

^{3}3x − 2x

^{3}2x

^{3}− 3x

**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`

**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`

**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

**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`

**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`

**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`

**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`

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 | 0.3 | 0.1 | 0.05 | 0.25 |

0·85

– 0·35

0·15

– 0·15

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

**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.

**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.

**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.

**Solve the following :**

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

**Solve the following:**

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

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).

**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 | 0.20 | 0.25 | 0.15 | 0.1 |

**Find the probability that**

**X is positive**

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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |

**Find the probability that**

**X is non-negative**

**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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |

**Find the probability that**

**X is odd**

**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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |

**Find the probability that**

**X is even**

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).

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.

**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.

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

A random variable X has the following probability distribution :

x = x | 0 | 1 | 2 | 3 | 7 | |||

P(X=x) | 0 | k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2} + k |

Determine (i) k

(ii) P(X> 6)

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

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).

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)

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)

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` |

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` |

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` |

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` |

**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.

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 ).

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)

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,"),(0 "otherwise".):}`

P(–1 < x < 1)

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)

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]` .

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).

## Solutions for Chapter 7: Probability Distributions

## 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 Variance of a Random Variable, Expected Value and Variance of a Random Variable, Random Variables and Its Probability Distributions, Types of Random Variables, Probability Distribution of Discrete Random Variables, Probability Distribution of a Continuous Random Variable.

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