#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrics

Chapter 1.3: Trigonometric Functions

Chapter 1.4: Pair of Lines

Chapter 1.5: Vectors and Three Dimensional Geometry

Chapter 1.6: Line and Plane

Chapter 1.7: Linear Programming Problems

Chapter 2.1: Differentiation

Chapter 2.2: Applications of Derivatives

Chapter 2.3: Indefinite Integration

Chapter 2.4: Definite Integration

Chapter 2.5: Application of Definite Integration

Chapter 2.6: Differential Equations

Chapter 2.7: Probability Distributions

Chapter 2.8: Binomial Distribution

## Chapter 3: Probability Distributions

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 3 Probability Distributions MCQ

#### 2 Mark each

Let the p.m.f. of a random variable X be P(x) = `(3 - x)/10`, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is ______

1

2

0

–1

c.d.f. of a discrete random variable X is

an identity function

a step function

an even function

an odd function

If X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______

`2/7`

`7/2`

1

`1/2`

A random variable X has the following probability distribution

X |
2 | 3 | 4 |

P(x) |
0.3 | 0.4 | 0.3 |

Then the variance of this distribution is

0.6

0.7

0.77

0.66

For the random variable X, if V(X) = 4, E(X) = 3, then E(x^{2}) is ______

9

13

12

7

If a d.r.v. X takes values 0, 1, 2, 3, … with 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`

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

a < b

a > b

a = b

a + b = 2

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(x/("n"("n" + 1))",", "for" x = 1"," 2"," 3"," .... "," "n"),(0",", "otherwise"):}`, then E(X) = ______

`"n" + 1/2`

`"n"/3 + 1/6`

`"n"/2 + 1/5`

`"n" + 1/3`

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for" x = 1"," 2"," 3","),(0",", "otherwise"):}` then E(X) = ______

`343/297`

`294/251`

`297/294`

`294/297`

If 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) is ______

`1/10`

`2/10`

`3/10`

`4/10`

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

then k = ______

`1/7`

`1/8`

`1/9`

`1/10`

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 3 Probability Distributions Very Short Answers

#### 1 Mark

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

Y |
−1 | 0 | 1 |

P(Y) |
0.6 | 0.1 | 0.2 |

Find mean for the following probability distribution.

X |
0 | 1 | 2 | 3 |

P(X = x) |
`1/6` | `1/3` | `1/3` | `1/6` |

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 |

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 3 Probability Distributions Short Answers I

#### 2 Marks

Find the expected value and variance of r.v. X whose p.m.f. is given below.

X |
1 | 2 | 3 |

P(X = x) |
`1/5` | `2/5` | `2/5` |

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

The probability distribution of X is as follows:

X |
0 | 1 | 2 | 3 | 4 |

P(X = x) |
0.1 | k | 2k | 2k | k |

Find k and P[X < 2]

**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 | 0.20 | 0.25 | 0.15 | 0.1 |

Find the probability that X is positive.

**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 | 0.20 | 0.25 | 0.15 | 0.1 |

Find the probability that X is odd.

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.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 3 Probability Distributions Short Answers II

#### 3 Marks

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

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 :

(i) k

(ii) P(X < 3)

(iii) P( X > 4)

Find the probability distribution of the number of doublets in three throws of a pair of dice

Find the mean and variance of the number randomly selected from 1 to 15

Let the p.m.f. of r.v. X be P(x) `{{:(((3 - x)/10",", "for" x = -1",", 0",", 1",", 2)),((0",", "otherwise")):}` Calculate E(X) and Var(X)

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

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 3 Probability Distributions Long Answers III

#### 4 Marks

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn

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

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).

**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.

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.

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

## Chapter 3: Probability Distributions

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 3 - Probability Distributions

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 3 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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