Mathematics · Class 12 Science
Ch 14Probability Distributions — Class 12 Mathematics, concept-first.
A random variable formalises an idea we already use informally: instead of listing every raw outcome of a random experiment, we are usually interested in a single number derived from that outcome — the number of heads when two coins are tossed, the sum shown by two dice, or the number of defective items in a sample.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Random Variable
A random variable is a real-valued function defined on the sample space of a random experiment: . Instead of tracking every raw outcome, it assigns a single real number to each outcome, letting us study numerical feature…
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Random variables
A random variable formalises an idea we already use informally: instead of listing every raw outcome of a random experiment, we are usually interested in a single number derived from that outcome — th…
+−Exercise 7.1i23 questions
- Q1Let X represent the difference between number of heads and number of tails obtained when a coin is tossed 6 times. What are possible values…Free
- Q2An 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…Free
- Q3State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (i) | X | 0 | 1 | 2 |…Free
- Q4State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (ii) | X | 0 | 1 | 2…Preview
- Q5State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (iii) | X | 0 | 1 | 2…Preview
- Q6State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (iv) | Z | 3 | 2 | 1…Preview
- Q7State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (v) | Y | –1 | 0 | 1…Preview
- Q8State which of the following are not the probability mass function of a random variable. Give reasons for your answer. (vi) | X | 0 | –1 | –…Preview
- Q9Find the probability distribution of (i) number of heads in two tosses of a coin.Preview
- Q10Find the probability distribution of (ii) Number of tails in the simultaneous tosses of three coins.Preview
- Q11Find the probability distribution of (iii) Number of heads in four tosses of a coin.Preview
- Q12Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 app…Preview
- Q13From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distributio…Preview
- Q14A 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 n…Preview
- Q15A random variable X has the following probability distribution : | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---|…Preview
- Q16Find 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…Preview
- Q17Find expected value and variance of X ,where X is number obtained on uppermost face when a fair die is thrown.Preview
- Q18Find the mean number of heads in three tosses of a fair coin.Preview
- Q19Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.Preview
- Q20Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers ob…Preview
- Q21Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X.Preview
- Q22A 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…Preview
- Q23In 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, a…Preview
Types of Random Variables
Having defined a random variable in general, the chapter now separates every random variable into exactly one of two families based on the nature of its range: discrete random variables, whose possibl…
+−Exercise 7.2i13 questions
- Q24Verify which of the following is p.d.f. of r.v. X : (i) f (x) = sin x, for 0 ≤ x ≤ π/2Free
- Q25Verify which of the following is p.d.f. of r.v. X : (ii) f (x) = x, for 0 ≤ x ≤ 1 and = 2 – x for 1 < x < 2Free
- Q26Verify which of the following is p.d.f. of r.v. X : (iii) f (x) = 2, for 0 ≤ x ≤ 1Free
- Q27The following is the p.d.f. of r.v. X : f (x) = x/8, for 0 < x < 4 and = 0 otherwise Find (a) P (x < 1.5 ) (b) P ( 1 < x < 2 ) (c) P( x > 2…Preview
- Q28It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by f (x) = x²/3, fo…Preview
- Q29Find k if the following function represent p.d.f. of r.v. X. (i) f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P (1/4 < x < 3/2).Preview
- Q30Find k if the following function represent p.d.f. of r.v. X. (ii) f (x) = kx (1 – x), for 0 < x < 1 and = 0 otherwise, Also find P (1/4 < x…Preview
- Q31Let 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 ≤…Preview
- Q32Suppose 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 probab…Preview
- Q33Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f. f (x) = k (4 – x²), for –2 ≤ x ≤ 2 and = 0 otherwise…Preview
- Q34The following is the p.d.f. of continuous r.v. f (x) = x/8, for 0 < x < 4 and = 0 otherwise. (i) Find expression for c.d.f. of X (ii) Find F…Preview
- Q35Given the p.d.f. of a continuous r.v. X , f (x) = x²/3, for –1 < x < 2 and = 0 otherwise Determine c.d.f. of X hence find P( x < 1) , P( x <…Preview
- Q36If 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).Preview
Discrete Random Variables
A random variable is said to be discrete if the number of its possible values is finite or countably infinite.
Continuous Random Variable
A random variable is said to be continuous if its possible values form an interval of real numbers rather than a countable list.
Probability Distribution of Discrete Random Variables
To see how a probability distribution is built, consider throwing two fair dice and noting the numbers on their upper faces. The sample space has equally likely ordered pairs, .
Probability Mass Function (p. m. f.)
Sometimes the probability of taking the value can be written as an explicit function of , valid for every possible value of ; such a function is called the probability mass function (p.m.f.) of the di…
Cumulative Distribution Function (c. d. f.)
The p.m.f. fully specifies a discrete random variable's probability distribution, but it is sometimes more convenient to work with a running total instead: the cumulative distribution function (c.d.f.…
Expected value and Variance of a random variable
Beyond the full probability distribution, it is often enough to summarise a random variable by a single representative number computed from that distribution.
Probability Distribution of a Continuous Random Variable
A continuous random variable behaves quite differently from a discrete one: its possible values fill an entire interval of real numbers, so there are uncountably infinitely many of them, and no single…
Probability Density Function (p. d. f.)
Let be a continuous random variable on the interval . A non-negative, integrable function is called the probability density function (p.d.f.) of if it satisfies three conditions: (1) for every ; (2) t…
Cumulative Distribution Functions (c. d. f.)
The cumulative distribution function of a continuous random variable extends the discrete definition in the natural way, replacing the sum by an integral.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 20 questionsHide questions20 questions
- Q1The probability distribution of a discrete random variable X is: | X = x | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(X = x) | k | 2k…Preview
- Q2The following is the p.d.f. (Probability Density Function) of a continuous random variable X: $f(x) = \dfrac{x}{32}$, $0 < x < 8$ $= 0$, oth…Preview
- Q3The expected value of the number of heads obtained when three fair coins are tossed simultaneously is (a) $1$ (b) $1.5$ (c) $0$ (d) $-1$Preview
- Q4Given the probability density function (p.d.f.) of a continuous random variable X as, $f(x) = \dfrac{x^2}{3}$, $-1 < x < 2$; $= 0$, otherwis…Preview
- Q5Let the p. m. f. of a random variable X be $P(x) = \dfrac{3-x}{10}$ for $x = -1, 0, 1, 2$, $= 0$ otherwise. Then $E(X)$ is ______. (a) 1 (b)…Preview
- Q6Find the variance and standard deviation of the random variable X whose probability distribution is given below: $x$: 0, 1, 2, 3; $P(X=x)$:…Preview
- Q7The probability distribution of a random variable X, the number of defects per 10 meters of a fabric is given by | x | 0 | 1 | 2 | 3 | 4 | |…Preview
- Q8Find expected value of the random variable X whose probability mass function is: X = x: 1, 2, 3; P(X = x): 1/5, 2/5, 2/5.Preview
- Q9If $f(x) = kx$, $0 < x < 2$; $= 0$, otherwise, is a probability density function of a random variable X, then find: i. Value of k. ii. $P(1…Preview
- Q10If the p.m.f. of a r.v. X is $P(x) = \dfrac{c}{x^3}$, for $x = 1, 2, 3$; $= 0$, otherwise, then $E(X) = $ ________. (a) $\dfrac{216}{251}$ (…Preview
- Q11Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.Preview
- Q12Let the probability mass function (p.m.f.) of a random variable X be $P(X=x) = {}^4C_x\left(\dfrac{5}{9}\right)^x \times \left(\dfrac{4}{9}\…Preview
- Q13Find the mean of number randomly selected from 1 to 15.Preview
- Q14For the following probability density function of a random variable X, find (a) $P(X<1)$ and (b) $P(|X|<1)$. $f(x) = \dfrac{x+2}{18}$; for $…Preview
- Q15If two coins are tossed simultaneously, write the probability distribution of the number of heads.Preview
- Q16Find $k$, if $f(x)=kx^2(1-x)$, for $0<x<1$, $=0$ otherwise, is the p.d.f. of random variable $X$.Preview
- Q17Suppose that $X$ is waiting time in minutes for a bus and its p.d.f. is given by: $f(x)=\dfrac{1}{5}$, for $0\le x\le 5$, $=0$, otherwise. F…Preview
- Q18Three coins are tossed simultaneously, $X$ is the number of heads. Find the expected value and variance of $X$.Preview
- Q19If the p.d.f. of a continuous r.v. $X$ is $f(x)=\dfrac{x+2}{18}$, for $-2<x<4$, $=0$, otherwise, then $P(|X|<1)=$ ____. (a) $\dfrac19$ (b) $…Preview
- Q20Find $E(X)$ and $V(X)$, where $X$ is the number obtained on uppermost face, when a fair die is thrown.Preview
More questions
32 Q+−Show 10 questionsHide questions10 questions
- Q37P.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 = (A) 1 (B) 1/2 (C)…Free
- Q38If 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) = (A) 2x − 3x² (…Free
- Q39If the p.d.f of a.c.r.v. X is f (x) = x²/18, for −3 < x < 3 and = 0, otherwise then P (| X | < 1) = (A) 1/27 (B) 1/28 (C) 1/29 (D) 1/26Free
- Q40If 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) = (A) 7/25…Preview
- Q41If p.m.f. of a d.r.v. X is P (X = x) = (5 Cx)/2⁵, for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise If a = P (X ≤ 2) and b = P (X ≥ 3), then E (X…Preview
- Q42If p.m.f. of a d.r.v. X is P (X = x) = x/(n (n + 1)), for x = 1, 2, 3, . . ., n and = 0, otherwise then E (X ) = (A) n/1 + 1/2 (B) n/3 + 1/6…Preview
- Q43If p.m.f. of a d.r.v. X is P (x) = c/x³ , for x = 1, 2, 3 and = 0, otherwise (elsewhere) then E (X ) = (A) 343/297 (B) 294/251 (C) 297/294 (…Preview
- Q44If the a d.r.v. X has the following probability distribution : | X | −2 | −1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---| | P (X = x) | 0…Preview
- Q45If the a d.r.v. X has the following probability distribution : | X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---| | P (X =…Preview
- Q46The expected value 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.…Preview
+−Show 22 questionsHide questions22 questions
- Q47Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (i) An economist is inte…Free
- Q48Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (ii) Amount of syrup pre…Free
- Q49Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (iii) The person on the…Free
- Q50Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (iv) 20 white rats are a…Preview
- Q51Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (v) A highway safety gro…Preview
- Q52The probability distribution of discrete r.v. X is as follows | X = x | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | P (X = x) |…Preview
- Q53The following probability distribution of r.v. X | X = x | –3 | –2 | –1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---|---| | P (X = x) | 0.…Preview
- Q54The p.m.f. of a r.v. X is given by P (X = x) = (5 Cx)/2⁵, for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise. Then show that P (X ≤ 2) = P (X ≥ 3).Preview
- Q55In 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.…Preview
- Q56A fair coin is tossed 4 times. Let X denotes the number of heads obtained write down the probability distribution of X. Also find the formul…Preview
- Q57Find 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…Preview
- Q58A random variable X has the following probability distribution. | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---|…Preview
- Q59The 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…Preview
- Q60Find the expected value, variance and standard deviation of r.v. X whose p.m.f. are given below. (i) | X = x | 1 | 2 | 3 | |---|---|---|---|…Preview
- Q61Find the expected value, variance and standard deviation of r.v. X whose p.m.f. are given below. (ii) | X = x | –1 | 0 | 1 | |---|---|---|--…Preview
- Q62Find the expected value, variance and standard deviation of r.v. X whose p.m.f. are given below. (iii) | X = x | 1 | 2 | 3 | . . . | n | |--…Preview
- Q63Find the expected value, variance and standard deviation of r.v. X whose p.m.f. are given below. (iv) | X = x | 0 | 1 | 2 | 3 | 4 | 5 | |---…Preview
- Q64A player tosses two coins he wins ` 10 if 2 heads appears , ` 5 if 1 head appears and ` 2 if no head appears. Find the expected winning amou…Preview
- Q65Let the p.m.f. of r.v. X be P (x) = (3 – x)/10 , for x = –1, 0, 1, 2 and = 0, otherwise Calculate E(X ) and Var (X ).Preview
- Q66Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f. f (x) = k (4 – x²) , for –2 ≤ x ≤ 2 and = 0 otherwis…Preview
- Q67The p.d.f. of 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).Preview
- Q68The p.d.f. of r.v. of X is given by f (x) = k/√x, for 0 < x < 4 and = 0, otherwise. Determine k . Determine c.d.f. of X and hence P (X ≤ 2)…Preview