#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 3: Probability Distributions

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.1

#### MCQ [1 Mark]

**Choose the correct alternative:**

The variance of a Binomial distribution is given by ______

np

pq

npq

`sqrt("npq")`

**Choose the correct alternative:**

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

x_{i} |
– 2 | – 1 | 0 | 1 | 2 |

p_{i} |
0.2 | 0.3 | 0.15 | 0.25 | 0.1 |

then F(– 3) = ______

0

1

0.2

0.15

**Choose the correct alternative:**

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

3.0

3.5

4.0

4.5

**Choose the correct alternative:**

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

x |
0 | 1 | 2 |

P(x) |
q^{2} |
2pq | p^{2} |

then Var(x) = ______

p

^{2}q

^{2}pq

2pq

**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

**Choose the correct alternative:**

If X ~ B`(20, 1/10)`, then E(x) = ______

2

5

4

3

**Choose the correct alternative:**

A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies ______

The trials are independent.

The probability of success remains the same in all trials.

The trials are independent but not the probability of success remains the same in all trials.

both trials are independent but not the probability of success remains the same in all trials.

**Choose the correct alternative:**

For the Poisson distribution ______

Mean = E(X) = m

Var(X) = m

Mean = E(X) = m and Var(X) = m

Mean = E(X) ≠ m and Var(X) = m

**Choose the correct alternative:**

A distance random variable X is said to have the Poisson distribution with parameter m if its p.m.f. is given by P(x) = `("e"^(-"m")"m"^"x")/("x"!)` the condition for m is ______

m > 0

m ≥ 0

m ≠ 1

m = 0

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.2

#### Fill in the blanks [1 Mark]

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

The values of continuous r.v. are generally obtained by ______

If X is discrete random variable takes the values x_{1}, x_{2}, x_{3}, … x_{n}, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______

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

In Binomial distribution, probability of success ______ from trial to trial

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

When n is very large and p is very small in the binomial distribution, then X follows the Poission distribution with prameter m = ______

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.3

#### [1 Mark]

**State whether the following statement 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

**State whether the following statement is True or False:**

If f(x) = `{:("k"x (1 - x)",", "for" 0 < x < 1),(= 0",", "otherwise"):}`

is the p.d.f. of a r.v. X, then k = 12

True

False

**State whether the following statement 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

**State whether the following statement is True or False:**

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

True

False

**State whether the following statement is True or False: **

Let X ~ B(n, p), then the mean or expected value of r.v. X is denoted by E(X). It is also denoted by E(X) and is given by µ = E(X) = npq

True

False

**State whether the following statement is True or False: **

A discrete random variable X is said to follow the Poisson distribution with parameter m ≥ 0 if its p.m.f. is given by P(X = x) = `("e"^(-"m")"m"^"x")/"x"`, x = 0, 1, 2, .....

True

False

**State whether the following statement is True or False:**

For the Binomial distribution, Mean E(X) = m and Variance = Var(X) = m

True

False

**State whether the following statement is True or False:**

If n is very large and p is very small then X follows Poisson distribution with n = mp

True

False

**State whether the following statement is True or False:**

The cumulative distribution function (c.d.f.) of a continuous random variable X is denoted by F and defined by

F(x) = `{:(0",", "for all" x ≤ "a"),( int_"a"^x f(x) "d"x",", "for all" x ≥ "a"):}`

True

False

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.4

#### Solve the following [3 Marks]

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

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

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

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.

Find the expected value and variance 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 |

Find the mean of 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

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

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

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

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

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.5

#### Solve the following [4 Marks]

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

Determine k

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

Determine P(X < 3)

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

Determine P(0 < X < 3)

A random variable X has the following probability distribution:

x |
1 | 2 | 3 | 4 | 5 | 6 | 7 |

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

Determine P(X > 4)

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)

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)

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)

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)

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).

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

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

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

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.

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.

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.

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.

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.

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.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.6

#### Activities [4 Marks]

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

x |
1 | 2 | 3 | 4 | 5 | 6 |

P(X = x) |
k | 2k | 3k | 4k | 5k | 6k |

Complete the following activity.

**Solution:** Since `sum"p"_"i"` = 1

k = `square`

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

x |
1 | 2 | 3 | 4 | 5 | 6 |

P(X = x) |
k | 2k | 3k | 4k | 5k | 6k |

Complete the following activity.

**Solution:** Since `sum"p"_"i"` = 1

P(X ≤ 4) = `square + square + square + square = square`

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

x |
1 | 2 | 3 | 4 | 5 | 6 |

P(X = x) |
k | 2k | 3k | 4k | 5k | 6k |

Complete the following activity.

**Solution:** Since `sum"p"_"i"` = 1

P(X ≥ 3) = `square - square - square = square`

Using the following activity, 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` |

**Solution:** µ = E(X) = `sum_("i" = 1)^3 x_"i""p"_"i"`

E(X) = `square + square + square = square`

Var(X) = `"E"("X"^2) - {"E"("X")}^2`

= `sum"X"_"i"^2"P"_"i" - [sum"X"_"i""P"_"i"]^2`

= `square - square`

= `square`

Let X ~ B(n, p). If n = 10 and E(X) = 5, using the following activity find p and Var(X)

**Solution:** E(X) = `square = 5 square "p" = square, "q" = square`

Var(X) = `square`

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

**Solution:** Let p = probability that bomb miss the target

∴ q = `square`, p = `square`, n = 5.

X ~ B`(5, square)`, P(x) = `""^"n""C"_x"P"^x"q"^("n" - x)`

P(X = 2) = `""^5"C"_2 square = square`

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, using the following activity find the value of m.

**Solution:** X : Follows Poisson distribution

∴ P(X) = `("e"^-"m" "m"^x)/(x!)`, P(X = 1) = 0.4 and P(X = 2) = 0.2

∴ P(X = 1) = `square` P(X = 2).

`("e"^-"m" "m"^x)/(1!) = square ("e"^-"m" "m"^2)/(2!)`,

`"e"^-"m" = square "e"^-"m" "m"/2`, m ≠ 0

∴ m = `square`

## Chapter 3: Probability Distributions

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 3 - Probability Distributions

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 3 (Probability Distributions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. SCERT Maharashtra Question Bank textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 3 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.

Using SCERT Maharashtra Question Bank 12th Board Exam solutions Probability Distributions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in SCERT Maharashtra Question Bank Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer SCERT Maharashtra Question Bank Textbook Solutions to score more in exam.

Get the free view of chapter 3 Probability Distributions 12th Board Exam extra questions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 and can use Shaalaa.com to keep it handy for your exam preparation