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RD Sharma solutions for Class 12 Mathematics chapter 32 - Mean and Variance of a Random Variable

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 32 : Mean and Variance of a Random Variable

Pages 14 - 16

Q 1 | Page 14

Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)

X : 3 2 1 0 −1
(X) : 0.3 0.2 0.4 0.1 0.05


(ii)

X : 0 1 2
P (X) : 0.6 0.4 0.2


(iii)

X : 0 1 2 3 4
P (X) : 0.1 0.5 0.2 0.1 0.1


(iv)

X : 0 1 2 3
P (X) : 0.3 0.2 0.4 0.1
Q 2 | Page 14

A random variable X has the following probability distribution:

Values of X : −2 −1 0 1 2 3
P (X) : 0.1 k 0.2 2k 0.3 k

Find the value of k.

Q 3 | Page 14

A random variable X has the following probability distribution:

Values of X : 0 1 2 3 4 5 6 7 8
P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a
 

Determine:
(i) The value of a
(ii) P (X < 3), P (X ≥ 3), P (0 < X < 5).

Q 4.1 | Page 14

The probability distribution function of a random variable X is given by

xi : 0 1 2
pi : 3c3 4c − 10c2 5c-1
 

where c > 0
Find: (i) c 

Q 4.2 | Page 14

The probability distribution function of a random variable X is given by

xi : 0 1 2
pi : 3c3 4c − 10c2 5c-1

where c > 0
Find:  (ii) P (X < 2) 

Q 4.3 | Page 14

The probability distribution function of a random variable X is given by

xi : 0 1 2
pi : 3c3 4c − 10c2 5c-1

where c > 0
Find: (iii) P (1 < X ≤ 2)

Q 5 | Page 14

Let X be a random variable which assumes values x1x2x3x4 such that 2P (X = x1) = 3P(X = x2) = P (X = x3) = 5 P (X = x4). Find the probability distribution of X.

 
Q 6 | Page 14

A random variable X takes the values 0, 1, 2 and 3 such that:
P (X = 0) = P (X > 0) = P (X < 0); P (X = −3) = P (X = −2) = P (X = −1); P (X = 1) = P (X = 2) = P (X = 3). Obtain the probability distribution of X.

Q 7 | Page 14

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 8 | Page 14

Find the probability distribution of the number of heads, when three coins are tossed.

 
Q 9 | Page 14

Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.

Q 10 | Page 14

A bag contains 4 red and 6 black balls. Three balls are drawn at random. Find the probability distribution of the number of red balls.

Q 11 | Page 14

Five defective mangoes are accidently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.

Q 12 | Page 15

Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?

Q 13 | Page 15

A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?

Q 14 | Page 15

Five defective bolts are accidently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.

Q 15 | Page 15

Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 16 | Page 15

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.

Q 17 | Page 15

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 18 | Page 15

Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls

Q 19 | Page 15

Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.

Q 20 | Page 15

From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let Xbe the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement.

 
Q 21 | Page 15

Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.

Q 22 | Page 15

An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.

Q 23 | Page 15

Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.

Q 24 | Page 15

A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.

Q 25 | Page 15

An urn contains 5 red and 2 black balls. Two balls are randomly selected. Let Xrepresent the number of black balls. What are the possible values of X. Is X a random variable?

Q 26 | Page 15

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

Q 27 | Page 15

From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.

Q 28 | Page 15

Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red balls drawn, then find the probability distribution of X.                         

Q 29.1 | Page 15

The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

(i) Determine the value of k

Q 29.2 | Page 15

The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

(ii) Determine P(X ≤ 2) and P(X > 2)

Q 29.3 | Page 15

The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

(iii) Find P(X ≤ 2) + P(X > 2)

 
Q 30 | Page 16

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}kx & , & if x = 0 or 1 \\ 2 kx & , & if x = 2 \\ k\left( 5 - x \right) & , & if x = 3 \text{ or } 4 \\ 0 & , & if x > 4\end{cases}\]

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Pages 4 - 44

Q 1.1 | Page 42

Find the mean and standard deviation of each of the following probability distributions:
(i)

xi : 2 3 4
pi : 0.2 0.5 0.3

 

Q 1.2 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(ii)

xi :  1 3 4 5
pi:  0.4 0.1 0.2 0.3

 

Q 1.3 | Page 42

Find the mean and standard deviation of each of the following probability distributions:


(iii)

xi : -5 -4 1 2
pi : \[\frac{1}{4}\] \[\frac{1}{8}\] \[\frac{1}{2}\] \[\frac{1}{8}\]
 
Q 1.4 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(iv)

xi : −1 0 1 2 3
pi : 0.3 0.1 0.1 0.3 0.2
Q 1.5 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(v) 

xi : 1 2 3 4
pi : 0.4 0.3 0.2 0.1
Q 1.6 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(vi)

xi: 0 1 3 5
pi :  0.2 0.5 0.2 0.1
Q 1.7 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(vii)

xi :  -2 -1 0 1 2
pi :  0.1 0.2 0.4 0.2 0.1
Q 1.8 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(viii)

xi :  -2 -1 0 1 2
pi :  0.05 0.45 0.20 0.25 0.05
Q 1.9 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(ix)

xi :  0 1 2 3 4 5
pi : 
\[\frac{1}{6}\]
\[\frac{5}{18}\]
\[\frac{2}{9}\]
\[\frac{1}{6}\]
\[\frac{1}{9}\]
\[\frac{1}{18}\]
Q 2.1 | Page 43

A discrete random variable X has the probability distribution given below:

X: 0.5 1 1.5 2
P(X): k k2 2k2 k

(i) Find the value of k.

Q 2.2 | Page 43

A discrete random variable X has the probability distribution given below:

X: 0.5 1 1.5 2
P(X): k k2 2k2 k

(ii) Determine the mean of the distribution.                

Q 3 | Page 43

Find the mean variance and standard deviation of the following probability distributio

xi : a b
pi : p q

where p + q = 1

Q 4 | Page 43

Find the mean and variance of the number of tails in three tosses of a coin.

Q 5 | Page 43

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.

Q 6 | Page 43

Find the mean, variance and standard deviation of the number of tails in three tosses of a coin.

 
Q 7 | Page 43

Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.

Q 8 | Page 43

A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.

 
Q 9 | Page 43

A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.

Q 10 | Page 4

A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.

 
Q 11 | Page 43

A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.

Q 12 | Page 43

A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.

Q 13 | Page 43

Two cards are selected at random from a box which contains five cards numbered 1, 1, 2, 2, and 3. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.

Q 14 | Page 43

A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.

Q 15 | Page 43

A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.

Q 16 | Page 43

In roulette, Figure, the wheel has 13 numbers 0, 1, 2, ...., 12 marked on equally spaced slots. A player sets Rs 10 on a given number. He receives Rs 100 from the organiser of the game if the ball comes to rest in this slot; otherwise he gets nothing. If X denotes the player's net gain/loss, find E (X).
Figure

Q 17 | Page 44

Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution.  

Q 18 | Page 44

An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, then find the mean and variance of X.      

Q 19 | Page 44

Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X

Q 20 | Page 44

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.

 

Pages 45 - 46

Q 1 | Page 45

Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:

Xxi: -2 -1 0 1
P(Xxi) :
\[\frac{1 - a}{4}\]
 
\[\frac{1 + 2a}{4}\]
\[\frac{1 - 2a}{4}\]
\[\frac{1 + a}{4}\]
Q 2 | Page 45

For what value of k the following distribution is a probability distribution?

X = xi : 0 1 2 3
P (X = xi) : 2k4 3k2 − 5k3 2k − 3k2 3k − 1
Q 3 | Page 45

If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.

Q 4 | Page 46

If the probability distribution of a random variable X is given by

X = xi : 1 2 3 4
P (X = xi) : 2k 4k 3k k

Write the value of k.

 
Q 5 | Page 45

Find the mean of the following probability distribution:

Xxi: 1 2 3
P(Xxi) :
\[\frac{1}{4}\]
 
\[\frac{1}{8}\]
\[\frac{5}{8}\]

 

Q 6 | Page 45

If the probability distribution of a random variable X is as given below:

X = xi : 1 2 3 4
P (X = xi) : c 2c 4c 4c

Write the value of P (X ≤ 2).

 
Q 7 | Page 45

A random variable has the following probability distribution:

X = xi : 1 2 3 4
P (X = xi) : k 2k 3k 4k

Write the value of P (X ≥ 3).

 

Pages 45 - 47

Q 1 | Page 45

If a random variable X has the following probability distribution:

X : 0 1 2 3 4 5 6 7 8
P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a

then the value of a is

(a) \[\frac{7}{81}\]

 

(b) \[\frac{5}{81}\]

 

(c) \[\frac{2}{81}\] 

 

(d)  \[\frac{1}{81}\]

 
Q 2 | Page 45

A random variable X has the following probability distribution:

X : 1 2 3 4 5 6 7 8
P (X) : 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05


For the events E = {X : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is

(a) 0.50

(b) 0.77

(c) 0.35

(d) 0.87

 
Q 3 | Page 45

A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is

(a) 0.1

(b) 0.2

(c) 0.3

(d) 0.4

 
Q 4 | Page 47

A random variable has the following probability distribution:

X = xi : 0 1 2 3 4 5 6 7
P (X = xi) : 0 2 p 2 p  3 p  p2 p2 p2 2 p 

The value of p is

(a) 1/10

(b) −1

(c) −1/10

(d) 1/5

 
Q 5 | Page 47

If X is a random-variable with probability distribution as given below:

X = xi : 0 1 2 3
P (X = xi) : k 3 k 3 k k

The value of k and its variance are


(a) 1/8, 22/27

(b) 1/8, 23/27

(c) 1/8, 24/27

(d) 1/8, 3/4

 
Q 6 | Page 47

Mark the correct alternative in the following question:
The probability distribution of a discrete random variable X is given below:

X: 2 3 4 5
P(X):
 

\[\frac{5}{k}\]
 

\[\frac{7}{k}\]
 

\[\frac{9}{k}\]
 


The value of k is

(a) 8   

 (b) 16 

  (c) 32     

  (d) 48

Q 7 | Page 47

Mark the correct alternative in the following question:
For the following probability distribution:

X: −4 −3 −2 −1 0
P(X): 0.1 0.2 0.3 0.2 0.2

The value of E(X) is

 

 

(a) 0  

    (b) −1 

   (c) −2 

  (d) −1.8

Q 8 | Page 47

Mark the correct alternative in the following question:

For the following probability distribution:
 

X: 1 2 3 4
P(X):
 

\[\frac{1}{10}\]
 

\[\frac{1}{5}\]
 

\[\frac{3}{10}\]
 

\[\frac{2}{5}\]


The value of E(X2) is

(a) 3     

     (b) 5   

      (c) 7   

    (d) 10

Q 9 | Page 47

Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is                                       

 

 

(a) E(X2)      

  (b) E(X2) + (E(X))2    

     (c) E(X2) - (E(X))2        

(d) \[\sqrt{E\left( X^2 \right) - \left( E\left( X \right) \right)^2}\]

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 32 - Mean and Variance of a Random Variable

RD Sharma solutions for Class 12 Maths chapter 32 (Mean and Variance of a Random Variable) 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 CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 32 Mean and Variance of a Random Variable are Properties of Conditional Probability, Introduction of Probability, Bernoulli Trials and Binomial Distribution, Variance of a Random Variable, Probability Examples and Solutions, Conditional Probability, Multiplication Theorem on Probability, Independent Events, Baye'S Theorem, Random Variables and Its Probability Distributions, Mean of a Random Variable.

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