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Mathematics · Class 12 Science

Ch 11Probability Distributions — Class 12 Mathematics, concept-first.

In Volume 2 of the Class 11 course, a sample space was built to completely describe the possible outcomes of a random experiment — for a coin toss, ; for a die, ; and so on.

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Random Variable — Discrete and Continuous

A random variable is a function defined on a sample space into the real numbers , such that the inverse image of every point, subset or interval of is an event in (Definition 11.1).

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Introduction

In Volume 2 of the Class 11 course, a sample space was built to completely describe the possible outcomes of a random experiment — for a coin toss, ; for a die, ; and so on.

11.2

Random Variable

Definition 11.1 (Random variable). A random variable is a function defined on a sample space into the real numbers , such that the inverse image of every point, subset or interval of is an event in —…

+Exercise 11.1i5 questions
  1. Q1Suppose $X$ is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable $X$…Free
  2. Q2In a pack of $52$ playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find…Free
  3. Q3An urn contains $5$ mangoes and $4$ apples. Three fruits are taken at random. If the number of apples taken is a random variable, then find…Preview
  4. Q4Two balls are chosen randomly from an urn containing $6$ red and $8$ black balls. Suppose that we win ₹15 for each red ball selected and we…Preview
  5. Q5A six-sided die is marked `2' on one face, `3' on two of its faces, and `4' on the remaining three faces. The die is thrown twice. If $X$ de…Preview
11.3

Types of Random Variable

This chapter restricts attention to two kinds of random variable:

11.3.1

Discrete random variables

Definition 11.2 (Discrete random variable). A random variable defined on a sample space into is a discrete random variable if its range is countable — it assumes only a finite or countably infinite nu…

11.3.2

Probability Mass Function

Definition 11.3 (Probability mass function). If is a discrete random variable taking the values , the function (also written ) defined by is the probability mass function (pmf) of .

11.3.3

Cumulative Distribution Function or Distribution Function

There are many situations where we want the probability that is at most some value , not just equal to it.

11.3.4

Cumulative Distribution Function from Probability Mass function

Both the pmf and the cdf of a discrete random variable carry the same information — the full probability distribution is determined by either one.

11.3.5

Probability Mass Function from Cumulative Distribution Function

The reverse conversion is just as direct: given the cdf , the pmf is recovered as the jump size of at each of its points of discontinuity.

+Exercise 11.2i7 questions
  1. Q1Three fair coins are tossed simultaneously. Find the probability mass function for the number of heads occurred.Free
  2. Q2A six-sided die is marked `1' on one face, `3' on two of its faces, and `5' on the remaining three faces. The die is thrown twice. If $X$ de…Free
  3. Q3Find the probability mass function and the cumulative distribution function of the number of girl children in families with $4$ children, as…Free
  4. Q4Suppose a discrete random variable can only take the values $0,1$ and $2$. The probability mass function is defined by $f(x)=\dfrac{x^2+1}{k…Preview
  5. Q5The cumulative distribution function of a discrete random variable is given by $$F(x)=\begin{cases}0 & -\infty<x<-1\\ 0.15 & -1\le x<0\\ 0.3…Preview
  6. Q6A random variable $X$ has the following probability mass function. $$\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5\\\hline f(x) & k^2 & 2k^2…Preview
  7. Q7The cumulative distribution function of a discrete random variable is given by $$F(x)=\begin{cases}0 & -\infty<x<0\\ \dfrac12 & 0\le x<1\\ \…Preview
11.4

Continuous Distributions

Some measurements — the current in a wire, the lifetime of a bulb — can take any value in an interval of real numbers, to any precision.

11.4.1

The definition of continuous random variable

Definition 11.5 (Continuous random variable). Let be a sample space and a random variable taking any value in a set . Then is a continuous random variable if for every .

11.4.2

Probability density function

Definition 11.6 (Probability density function). A non-negative real-valued function is a probability density function (pdf) of a continuous random variable if, for every in the range of , — i.e.

11.4.3

Distribution function (Cumulative distribution function)

Definition 11.7 (Cumulative distribution function, continuous case). For a continuous random variable with pdf , the distribution function is

11.4.3.1

Properties of distribution function

For a continuous random variable , the cdf satisfies six standing properties, closely paralleling — but not identical to — the discrete case: (i) for every . (ii) is real-valued and non-decreasing: .

11.4.4

Distribution function from Probability density function

Both and carry the full probability information of a continuous , and either determines the other. Given the pdf , the cdf is built by integrating piece by piece over each interval on which has a diff…

11.4.5

Probability density function from Probability distribution function

Working rule. If is the cdf of a continuous random variable , its pdf is recovered by differentiation:

+Exercise 11.3i6 questions
  1. Q1The probability density function of $X$ is given by $f(x)=\begin{cases}kxe^{-2x} & x>0\\ 0 & x\le0\end{cases}$. Find the value of $k$.Free
  2. Q2The probability density function of $X$ is $f(x)=\begin{cases}x & 0<x<1\\ 2-x & 1\le x<2\\ 0 & \text{otherwise}\end{cases}$. Find (i) $P(0.2…Free
  3. Q3Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of $200$ litres and a maximum of $600$ litres, with prob…Preview
  4. Q4The probability density function of $X$ is given by $f(x)=\begin{cases}ke^{-x/3} & x>0\\ 0 & x\le0\end{cases}$. Find (i) the value of $k$ (i…Preview
  5. Q5If $X$ is the random variable with probability density function $f(x)$ given by $$f(x)=\begin{cases}x+1 & -1\le x<0\\ -x+1 & 0\le x<1\\ 0 &…Preview
  6. Q6If $X$ is the random variable with distribution function $F(x)$ given by $$F(x)=\begin{cases}0 & -\infty<x<0\\ \dfrac12(x^2+x) & 0\le x<1\\…Preview
11.5

Mathematical Expectation

One of the most important characteristics of a random variable is its expectation — synonyms include expected value, mean, and first moment.

11.5.1

Mean

Definition 11.8 (Mean). For a random variable with pmf/pdf , the expected value (mean), written or , is need not itself be a value can take (e.g.

11.5.2

Variance

Definition 11.9 (Variance). The variance of , written , , or , is It measures how the data spread out from the mean — the average of the squared deviations from .

11.5.3

Properties of Mathematical expectation and variance

Three linearity/scaling laws hold for any random variable and constants .

+Exercise 11.4i8 questions
  1. Q1For the random variable $X$ with the given probability mass function below, find the mean and variance. (i) $f(x)=\dfrac1{10}$ for $x=2,5$ a…Free
  2. Q2Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let $X$ be the number of…Free
  3. Q3If $\mu$ and $\sigma^2$ are the mean and variance of the discrete random variable $X$, and $E(X+3)=10$ and $E\big((X+3)^2\big)=116$, find $\…Free
  4. Q4Four fair coins are tossed once. Find the probability mass function, mean and variance for the number of heads occurred.Preview
  5. Q5A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station. Let $X$ den…Preview
  6. Q6The time to failure (in thousands of hours) of an electronic equipment used in a manufactured computer has the density function $$f(x)=\begi…Preview
  7. Q7The probability density function of the random variable $X$ is given by $$f(x)=\begin{cases}16xe^{-4x} & x>0\\ 0 & x\le0\end{cases}$$ Find t…Preview
  8. Q8A lottery with $600$ tickets gives one prize of ₹200, four prizes of ₹100, and six prizes of ₹50. If the ticket costs ₹2, find the expected…Preview
11.6

Theoretical Distributions: Some Special Discrete Distributions

Having developed the general pmf/cdf/mean/variance machinery, this section studies four named discrete distributions of special importance, in increasing order of richness: the one-point distribution…

11.6.1

The One point distribution

A random variable has a one-point distribution if all its probability mass sits at a single value : Its cdf is the unit step for and for .

11.6.2

The Two point distribution

(a) Unsymmetrical case. has a two-point distribution if it takes two values with with cdf for , for , and for .

11.6.3

The Bernoulli distribution

Named for the Swiss mathematician Jacob Bernoulli (1654-1705), a Bernoulli trial is a random experiment whose outcome is classified into exactly one of two mutually exclusive, exhaustive categories: s…

11.6.4

The Binomial Distribution

29 Q

The binomial distribution applies to repeated independent trials with only two possible outcomes each — heads/tails, defective/good, and so on — the probability of each sequence computed via the multi…

+Exercise 11.5i9 questions
  1. Q1Compute $P(X=k)$ for the binomial distribution $B(n,p)$ where (i) $n=6,\ p=\dfrac13,\ k=3$ (ii) $n=10,\ p=\dfrac15,\ k=4$ (iii) $n=9,\ p=\df…Free
  2. Q2The probability that Mr. Q hits a target at any trial is $\dfrac14$. Suppose he tries at the target $10$ times. Find the probability that he…Free
  3. Q3Using the binomial distribution, find the mean and variance of $X$ for the following experiments. (i) A fair coin is tossed $100$ times, and…Free
  4. Q4The probability that a certain kind of component will survive an electrical test is $\dfrac34$. Find the probability that exactly $3$ of the…Preview
  5. Q5A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the devi…Preview
  6. Q6If the probability that a fluorescent light has a useful life of at least $600$ hours is $0.9$, find the probabilities that among $12$ such…Preview
  7. Q7The mean and standard deviation of a binomial variate $X$ are respectively $6$ and $2$. Find (i) the probability mass function (ii) $P(X=3)$…Preview
  8. Q8If $X\sim B(n,p)$ is such that $4P(X=4)=P(X=2)$ and $n=6$, find the distribution, mean and standard deviation of $X$.Preview
  9. Q9In a binomial distribution consisting of $5$ independent trials, the probabilities of $1$ and $2$ successes are $0.4096$ and $0.2048$ respec…Preview
+Exercise 11.6i20 questions
  1. Q1Let $X$ be a random variable with probability density function $$f(x)=\begin{cases}\dfrac{2}{x^3} & x\ge1\\ 0 & x<1\end{cases}$$ Which of th…Free
  2. Q2A rod of length $2l$ is broken into two pieces at random. The probability density function of the shorter of the two pieces is $$f(x)=\begin…Free
  3. Q3Consider a game where the player tosses a six-sided fair die. If the face that comes up is $6$, the player wins ₹36; otherwise he loses ₹$k^…Free
  4. Q4A pair of dice numbered $1,2,3,4,5,6$ of a six-sided die and $1,2,3,4$ of a four-sided die is rolled and the sum is determined. Let the rand…Preview
  5. Q5A random variable $X$ has a binomial distribution with $n=25$ and $p=0.8$. Then the standard deviation of $X$ is (1) $6$ (2) $4$ (3) $3$ (4)…Preview
  6. Q6Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed $n$ times. Then the poss…Preview
  7. Q7If the function $f(x)=\dfrac1{12}$ for $a<x<b$ represents a probability density function of a continuous random variable $X$, then which of…Preview
  8. Q8Four buses carrying $160$ students from the same school arrive at a football stadium. The buses carry, respectively, $42,36,34$ and $48$ stu…Preview
  9. Q9Two coins are to be flipped. The first coin will land on heads with probability $0.6$, the second with probability $0.5$. Assume that the re…Preview
  10. Q10On a multiple-choice exam with $3$ possible answers for each of the $5$ questions, the probability that a student will get $4$ or more corre…Preview
  11. Q11If $P(X=0)=1-P(X=1)$ and $E(X)=3\,\text{Var}(X)$, then $P(X=0)$ is (1) $\dfrac23$ (2) $\dfrac25$ (3) $\dfrac15$ (4) $\dfrac13$.Preview
  12. Q12If $X$ is a binomial random variable with expected value $6$ and variance $2.4$, then $P(X=5)$ is (1) $\dbinom{10}5\left(\dfrac35\right)^6\l…Preview
  13. Q13The random variable $X$ has the probability density function $$f(x)=\begin{cases}ax+b & 0<x<1\\ 0 & \text{otherwise}\end{cases}$$ and $E(X)=…Preview
  14. Q14Suppose that $X$ takes on one of the values $0,1$ and $2$. If, for some constant $k$, $P(X=i)=kP(X=i-1)$ for $i=1,2$ and $P(X=0)=\dfrac17$,…Preview
  15. Q15Which of the following is a discrete random variable? I. The number of cars crossing a particular signal in a day. II. The number of custome…Preview
  16. Q16If $f(x)=\begin{cases}2x & 0\le x\le a\\ 0 & \text{otherwise}\end{cases}$ is a probability density function of a random variable, then the v…Preview
  17. Q17The probability mass function of a random variable is defined as $$\begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2\\\hline f(x) & k & 2k & 3k…Preview
  18. Q18Let $X$ have a Bernoulli distribution with mean $0.4$. Then the variance of $(2X-3)$ is (1) $0.24$ (2) $0.48$ (3) $0.6$ (4) $0.96$.Preview
  19. Q19If, in $6$ trials, $X$ is a binomial variable which follows the relation $9P(X=4)=P(X=2)$, then the probability of success is (1) $0.125$ (2…Preview
  20. Q20A computer salesperson knows from past experience that he sells a computer to one in every twenty customers who enter the showroom. What is…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 50 questions50 questions
  1. Q1If $E(X+C)=8$ and $E(X-C)=12$ then the value of $C$ is : (a) $-2$ (b) $4$ (c) $-4$ (d) $2$Preview
  2. Q2A random variable X has the following p.d.f. | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---| | P(X=x) | 0 | k |…Preview
  3. Q3The marks secured by 400 students in a Mathematics test were normally distributed with mean 65. If 120 students got marks above 85, the numb…Preview
  4. Q4In 16 throws of a die getting an even number is considered a success, then the variance of the successes is (a) $4$ (b) $6$ (c) $2$ (d) $256…Preview
  5. Q5Two cards are drawn with replacement from a well shuffled deck of 52 cards. Find the mean and variance for the number of aces.Preview
  6. Q620% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly…Preview
  7. Q7The mean score of 1000 students for an examination is 34 and S.D. is 16. (i) How many candidates can be expected to obtain marks between 30…Preview
  8. Q8The random variable X follows normal distribution $f(x) = ce^{-\frac{1}{2}\frac{(x-100)^2}{25}}$. Then the value of c is : (a) $\sqrt{2\pi}$…Preview
  9. Q9A random variable X has the following probability distribution : | X | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | P(X=x) | $\df…Preview
  10. Q10The mean of a binomial distribution is 5 and its standard deviation is 2. Then the values of n and p are : (a) $\left(\dfrac{4}{5}, 25\right…Preview
  11. Q11A continuous random variable X has p.d.f. $f(x)$, then : (a) $0 \le f(x) \le 1$ (b) $f(x) \ge 0$ (c) $f(x) \le 1$ (d) $0 < f(x) < 1$Preview
  12. Q12(i) Let Z be a standard normal variate. Find the value of c if $P(Z < c) = 0.05$. Here $P[0 < Z < 1.65] = 0.45$ (ii) The difference between…Preview
  13. Q13A die is tossed twice. A success is getting an odd number on a toss. Find the mean and the variance of the probability distribution of the n…Preview
  14. Q14Verify $f(x) = \begin{cases}30x^4 e^{-6x^5} & ; x > 0\\0 & ; \text{Otherwise}\end{cases}$ for p.d.f. If $f(x)$ is a p.d.f. then find F(1).Preview
  15. Q15A random variable X has the following probability mass function : | X | $-2$ | $3$ | $1$ | |---|---|---|---| | $P(X=x)$ | $\dfrac{\lambda}{6…Preview
  16. Q16If, in a Poisson distribution $P(X=0) = k$ then the variance is : (a) $e^{\lambda}$ (b) $\log \dfrac{1}{k}$ (c) $\dfrac{1}{k}$ (d) $\log k$Preview
  17. Q17The mean of a binomial distribution is 5 and its standard deviation is 2. Then the value of $n$ and $p$ are : (a) $\left(\dfrac{1}{5}, 25\ri…Preview
  18. Q18Which of the following is not true regarding the normal distribution ? (a) the point of inflection are at $X = \mu \pm \sigma$ (b) skewness…Preview
  19. Q19Two unbiased dice are thrown together at random. Find the expected value of the total number of points shown up.Preview
  20. Q20If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches, find the height below which…Preview
  21. Q21If the number of incoming buses per minute at a bus terminus is a random variable having a Poisson distribution with $\lambda = 0.9$, find t…Preview
  22. Q22Variance of the random variable $X$ is $4$. Its mean is $2$. Then $E(X^2)$ is : (a) $6$ (b) $8$ (c) $2$ (d) $4$Preview
  23. Q23In a Poisson distribution if $P(X = 2) = P(X = 3)$ then, the value of its parameter $\lambda$ is : (a) $3$ (b) $0$ (c) $6$ (d) $2$Preview
  24. Q24Prove that $F(3) = 1 - e^{-9}$ if the probability density function $f(x)$ is defined as $f(x) = \begin{cases} 3e^{-3x}, & x>0 \\ 0, & x \le…Preview
  25. Q25A die is thrown 120 times and getting 1 or 5 is considered a success. Find the mean and variance of the number of successes.Preview
  26. Q26(a) The mean score of 1000 students for an examination is 34 and the standard deviation is 16. Determine the limit of the marks of the centr…Preview
  27. Q27A random variable X has binomial distribution with $n=25$ and $p=0.8$, then the standard deviation of X is : (a) $2$ (b) $6$ (c) $4$ (d) $3$Preview
  28. Q28A random variable X has the following probability mass function : | X | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | P(X=x) | k |…Preview
  29. Q29Let X be a continuous random variable and $f(x)$ is defined as : $$f(x)=\begin{cases}kx(1-x)^{10}, & 0<x<1\\0, & \text{otherwise}\end{cases}…Preview
  30. Q30If $f(x)=\begin{cases}2x, & 0\le x\le a\\0, & \text{otherwise}\end{cases}$ is a probability density function of a random variable, then the…Preview
  31. Q31Which one of the following is not true in the case of discrete random variable X ? (a) $\displaystyle\lim_{x\to\infty}F(x)=F(\infty)=1$ (b)…Preview
  32. Q32A random variable X has the following probability mass function. | x | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | $f(x)$ | k |…Preview
  33. Q33X is the number of tails occurred when three fair coins are tossed simultaneously. Find the values of the random variable X and number of po…Preview
  34. Q34(a) The distribution function of a continuous random variable X is : $$F(x)=\begin{cases}0, & x<1\\\dfrac{x-1}{4}, & 1\le x\le5\\1, & x>5\en…Preview
  35. Q35Suppose that X takes on one of the values $0, 1, 2$. If for some constant k, $P(X=i)=kP(X=i-1)$ for $i=1, 2$ and $P(X=0)=\dfrac17$, then the…Preview
  36. Q36The value of $\text{Var}(3)$ is : (a) $0$ (b) $3$ (c) $\text{Var}(3)$ (d) $9$Preview
  37. Q37The random variable X has binomial distribution with $n=25$ and $p=0.8$ then standard deviation of X is : (a) $3$ (b) $6$ (c) $2$ (d) $4$Preview
  38. Q38For the random variable X with the given probability mass function $f(x)=\begin{cases}2(x-1), & 1<x<2\\0, & \text{otherwise}\end{cases}$. Fi…Preview
  39. Q39Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.Preview
  40. Q40(a) A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X deno…Preview
  41. Q41Suppose that X takes on one of the values 0, 1 and 2. If for some constant k, $P(X=i)=k\,P(X=i-1)$ for $i=1,2$ and $P(X=0)=\dfrac17$, then t…Preview
  42. Q42If in 6 trials, X is a binomial variable which follows the relation $9P(X=4)=P(X=2)$, then the probability of success is : (a) $0.375$ (b) $…Preview
  43. Q43Find the constant C such that the function $f(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}$ is a density function of X.Preview
  44. Q44A lottery with 600 tickets gives one prize of ₹200, four prizes of ₹100 and six prizes of ₹50. If the ticket cost is ₹2, find the expected p…Preview
  45. Q45(a) During war, 1 ship out of 9 was sunk on an average in making a certain voyage. What was the probability that : (i) Exactly 3 out of a co…Preview
  46. Q46If the function $f(x)=\dfrac{1}{12}$ for $a<x<b$, represents a probability density function of a continuous random variable X, then which of…Preview
  47. Q47A rod of length $2l$ is broken into two pieces at random. The probability density function of the shorter of the two pieces is $f(x)=\begin{…Preview
  48. Q48If $\sqrt{\text{Var}(X)}=\dfrac12$ then, find the value of $\text{Var}(2X+3)$.Preview
  49. Q49If $X\sim B(n, p)$ such that $4P(X=4)=P(X=2)$ and $n=6$, find the distribution, mean and Standard Deviation of X.Preview
  50. Q50(a) A random variable X has the following probability mass function. | x | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | f(x) | k…Preview