Chapters
Chapter 2: Functions
Chapter 3: Binary Operations
Chapter 4: Inverse Trigonometric Functions
Chapter 5: Algebra of Matrices
Chapter 6: Determinants
Chapter 7: Adjoint and Inverse of a Matrix
Chapter 8: Solution of Simultaneous Linear Equations
Chapter 9: Continuity
Chapter 10: Differentiability
Chapter 11: Differentiation
Chapter 12: Higher Order Derivatives
Chapter 13: Derivative as a Rate Measurer
Chapter 14: Differentials, Errors and Approximations
Chapter 15: Mean Value Theorems
Chapter 16: Tangents and Normals
Chapter 17: Increasing and Decreasing Functions
Chapter 18: Maxima and Minima
Chapter 19: Indefinite Integrals
Chapter 20: Definite Integrals
Chapter 21: Areas of Bounded Regions
Chapter 22: Differential Equations
Chapter 23: Algebra of Vectors
Chapter 24: Scalar Or Dot Product
Chapter 25: Vector or Cross Product
Chapter 26: Scalar Triple Product
Chapter 27: Direction Cosines and Direction Ratios
Chapter 28: Straight Line in Space
Chapter 29: The Plane
Chapter 30: Linear programming
Chapter 31: Probability
Chapter 32: Mean and Variance of a Random Variable
Chapter 33: Binomial Distribution
RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (201819 Session)
Chapter 32: Mean and Variance of a Random Variable
Chapter 32: Mean and Variance of a Random Variable solutions [Pages 14  16]
Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)
X :  3  2  1  0  −1 
P (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 
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.
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).
The probability distribution function of a random variable X is given by
x_{i} :  0  1  2 
p_{i} :  3c^{3}  4c − 10c^{2}  5c1 
where c > 0
Find: (i) c
The probability distribution function of a random variable X is given by
x_{i} :  0  1  2 
p_{i} :  3c^{3}  4c − 10c^{2}  5c1 
where c > 0
Find: (ii) P (X < 2)
The probability distribution function of a random variable X is given by
x_{i} :  0  1  2 
p_{i} :  3c^{3}  4c − 10c^{2}  5c1 
where c > 0
Find: (iii) P (1 < X ≤ 2)
Let X be a random variable which assumes values x_{1}, x_{2}, x_{3}, x_{4} such that 2P (X = x_{1}) = 3P(X = x_{2}) = P (X = x_{3}) = 5 P (X = x_{4}). Find the probability distribution of X.
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.
Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of the number of heads, when three coins are tossed.
Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.
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.
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.
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?
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?
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.
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.
Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
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
Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
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.
Three cards are drawn successively with replacement from a wellshuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.
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.
Two cards are drawn simultaneously from a wellshuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.
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.
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?
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?
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.
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.
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
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)
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)
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 x number of colleges. It is given that
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.
Chapter 32: Mean and Variance of a Random Variable solutions [Pages 4  44]
Find the mean and standard deviation of each of the following probability distributions:
(i)
x_{i} :  2  3  4 
p_{i} :  0.2  0.5  0.3 
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 
Find the mean and standard deviation of each of the following probability distributions:
(iii)
x_{i} :  5  4  1  2 
p_{i} :  \[\frac{1}{4}\]  \[\frac{1}{8}\]  \[\frac{1}{2}\]  \[\frac{1}{8}\] 
Find the mean and standard deviation of each of the following probability distributions:
(iv)
x_{i} :  −1  0  1  2  3 
p_{i} :  0.3  0.1  0.1  0.3  0.2 
Find the mean and standard deviation of each of the following probability distributions:
(v)
x_{i} :  1  2  3  4 
p_{i}_{ }:  0.4  0.3  0.2  0.1 
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 
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 
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 
Find the mean and standard deviation of each of the following probability distributions:
(ix)
xi :  0  1  2  3  4  5 
pi : 
\[\frac{1}{6}\]

\[\frac{5}{18}\]

\[\frac{2}{9}\]

\[\frac{1}{6}\]

\[\frac{1}{9}\]

\[\frac{1}{18}\]

A discrete random variable X has the probability distribution given below:
X:  0.5  1  1.5  2 
P(X):  k  k^{2}  2k^{2}  k 
(i) Find the value of k.
A discrete random variable X has the probability distribution given below:
X:  0.5  1  1.5  2 
P(X):  k  k^{2}  2k^{2}  k 
(ii) Determine the mean of the distribution.
Find the mean variance and standard deviation of the following probability distributio
x_{i}_{ }:  a  b 
p_{i} :  p  q 
where p + q = 1
Find the mean and variance of the number of tails in three tosses of a coin.
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
Find the mean, variance and standard deviation of the number of tails in three tosses of a coin.
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.
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.
A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.
A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.
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.
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.
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.
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
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.
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
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.
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.
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.
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.
Chapter 32: Mean and Variance of a Random Variable solutions [Pages 45  46]
Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:
X= x_{i}:  2  1  0  1 
P(X= x_{i}) : 
\[\frac{1  a}{4}\]

\[\frac{1 + 2a}{4}\]

\[\frac{1  2a}{4}\]

\[\frac{1 + a}{4}\]

For what value of k the following distribution is a probability distribution?
X = x_{i} :  0  1  2  3 
P (X = x_{i}) :  2k^{4}  3k^{2} − 5k^{3}  2k − 3k^{2}  3k − 1 
If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.
If the probability distribution of a random variable X is given by
X = x_{i} :  1  2  3  4 
P (X = x_{i}) :  2k  4k  3k  k 
Write the value of k.
Find the mean of the following probability distribution:
X= x_{i}:  1  2  3 
P(X= x_{i}) : 
\[\frac{1}{4}\]

\[\frac{1}{8}\]

\[\frac{5}{8}\]

If the probability distribution of a random variable X is as given below:
X = x_{i} :  1  2  3  4 
P (X = x_{i}) :  c  2c  4c  4c 
Write the value of P (X ≤ 2).
A random variable has the following probability distribution:
X = x_{i} :  1  2  3  4 
P (X = x_{i}) :  k  2k  3k  4k 
Write the value of P (X ≥ 3).
Chapter 32: Mean and Variance of a Random Variable solutions [Pages 45  47]
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) \[\frac{1}{81}\]
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
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
A random variable has the following probability distribution:
X = x_{i} :  0  1  2  3  4  5  6  7 
P (X = x_{i}) :  0  2 p  2 p  3 p  p^{2}  2 p^{2}  7 p^{2}  2 p 
The value of p is
(a) 1/10
(b) −1
(c) −1/10
(d) 1/5
If X is a randomvariable with probability distribution as given below:
X = x_{i} :  0  1  2  3 
P (X = x_{i}) :  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
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
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
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(X^{2}) is
(a) 3
(b) 5
(c) 7
(d) 10
Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is
(a) E(X^{2})
(b) E(X^{2}) + (E(X))^{2}
(c) E(X^{2})  (E(X))^{2}
(d) \[\sqrt{E\left( X^2 \right)  \left( E\left( X \right) \right)^2}\]
Chapter 32: Mean and Variance of a Random Variable
RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (201819 Session)
Textbook solutions for Class 12
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, stepbystep 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) (201819 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, Mean of a Random Variable, Random Variables and Its Probability Distributions, Baye'S Theorem, Independent Events, Multiplication Theorem on Probability, Conditional Probability, Variance of a Random Variable, Probability Examples and Solutions.
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