Skip to content
← Business Mathematics and Statistics

Business Mathematics and Statistics · Class 12 Commerce

Ch 6Random Variable and Mathematical Expectation — Class 12 Business Mathematics and Statistics, concept-first.

A random variable is a rule (a function) that assigns a single real number to every possible outcome of a random experiment. It is the bridge between the outcomes of an experiment — which may be words, objects, or events — and the numbers we need in order to apply arithmetic, algebra, and later calculus to probability.…

41

Q&A

6

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Cumulative Distribution Function

The cumulative distribution function gives the probability that a random variable takes a value up to and including .

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Meaning and Types of a Random Variable

A random variable is a rule (a function) that assigns a single real number to every possible outcome of a random experiment.

2

Probability Mass Function of a Discrete Random Variable

When a random variable is discrete, the way its probabilities are distributed over its possible values is described by its probability mass function, written , or simply .

3

Probability Density Function of a Continuous Random Variable

A continuous random variable cannot be described by listing probabilities for individual values, because for every single — there are simply too many (uncountably many) values in any interval for any…

4

Cumulative Distribution Function

The cumulative distribution function, denoted , gives the probability that a random variable takes a value less than or equal to a given number :

5

Mathematical Expectation of a Random Variable

The mathematical expectation (or expected value) of a discrete random variable , denoted , is defined as:

6

Variance and Standard Deviation of a Random Variable

Mathematical expectation tells us the centre of a distribution, but says nothing about how spread out or unpredictable the values are around that centre.

7

Application: Expectation and Variance in Business Decision-Making

The real payoff of everything built up in this chapter is that mathematical expectation and variance turn an uncertain business situation into two clear numbers a decision can be based on.

Exercises

+Show 6 questions6 questions
  1. Q8Distinguish between the probability mass function (p.m.f.) and the cumulative distribution function (c.d.f.) of a discrete random variable.…Free
  2. Q9If $E(X) = 5$ and $Var(X) = 4$ for a random variable $X$, find (i) $E(2X+3)$ and (ii) $Var(3X-2)$.Free
  3. Q10If $SD(X) = 3$, find $SD(4X+5)$.Preview
  4. Q11Two random variables $X$ and $Y$ have $E(X) = 8$ and $E(Y) = 5$. Find (i) $E(X+Y)$ and (ii) $E(2X - 3Y + 7)$.Preview
  5. Q12A company is comparing two investment proposals for the coming year. Proposal A gives a profit of ₹80,000 with probability 0.4, ₹20,000 with…Preview
  6. Q13The number of defective bulbs $X$ found when 3 bulbs are randomly checked from a large batch has the probability distribution: $P(X=0)=0.512…Preview

Sample & Board Papers

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

+Show 28 questions28 questions
  1. Q1Demand of products per day for three days are $21, 19, 22$ and their respective probabilities are $0.29, 0.40, 0.35$. Profit per unit is $0.…Preview
  2. Q2The probability function of a random variable is defined as : | $X = x$ | $-1$ | $-2$ | $0$ | $1$ | $2$ | | --- | --- | --- | --- | --- | --…Preview
  3. Q3What are the properties of continuous random variable ?Preview
  4. Q4In a business venture, a man can make a profit of $\text{₹ } 2{,}000$ with a probability of $0.4$ or have a loss of $\text{₹ } 1{,}000$ with…Preview
  5. Q5(a) The amount of bread (in hundreds of pounds) $x$ that a certain bakery is able to sell in a day is found to be a numerical valued random…Preview
  6. Q6If c is a constant in a continuous probability distribution, then $p(x=c)$ is always equal to ______. (a) negative (b) zero (c) does not exi…Preview
  7. Q7The probability function of a random variable is defined as : | $X = x$ | $-1$ | $-2$ | $0$ | $1$ | $2$ | | --- | --- | --- | --- | --- | --…Preview
  8. Q8Two unbiased dice are thrown simultaneously and sum of the upturned faces considered as random variable. Construct a probability mass functi…Preview
  9. Q9Let X be a random variable defining number of students getting A grade. Find the expected value of X from the given table. | $X = x$ | $0$ |…Preview
  10. Q10What are the properties of Mathematical expectation ?Preview
  11. Q11A person tosses a coin and is to receive ₹ 6 for a head and is to pay ₹ 3 for a tail. Find the expectation and variance of his gains.Preview
  12. Q12$E[X-E(X)]$ is equal to : (a) $0$ (b) $E(X)$ (c) $E(X)-X$ (d) $V(X)$Preview
  13. Q13If $p(x)=\dfrac{1}{10}$, $x=10$ then $E(X)$ is : (a) $1$ (b) Zero (c) $-1$ (d) $\dfrac{6}{8}$Preview
  14. Q14The following information is the probability distribution of successes. | No. of successes $X=x$ | 0 | 1 | 2 | | --- | --- | --- | --- | | P…Preview
  15. Q15Consider a random variable X with probability density function, $f(x)=\begin{cases}4x^3, & \text{if } 0<x<1 \\ 0, & \text{otherwise}\end{cas…Preview
  16. Q16Given $E(X)=5$ and $E(Y)=-2$ then $E(X-Y)$ is : (a) $3$ (b) $7$ (c) $5$ (d) $-2$Preview
  17. Q17In a discrete probability distribution the sum of all the probabilities is always equal to : (a) one (b) minimum (c) zero (d) maximumPreview
  18. Q18Prove that if $E(X)=0$, then $V(X)=E(X^2)$Preview
  19. Q19A continuous random variable X has the following distribution function. $$F(x)=\begin{cases} 0 & \text{if } x\le 1 \\ k(x-1)^4 & \text{if }…Preview
  20. Q20(a) A continuous random variable X has the following probability function. | $X=x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | --- | --- | --- | ---…Preview
  21. Q21Probability which explains $x$ is equal to or less than particular value is classified as : (a) marginal probability (b) discrete probabilit…Preview
  22. Q22$E[X-E(X)]^{2}$ is : (a) $V(X)$ (b) $E(X)$ (c) $S.D(X)$ (d) $E(X^{2})$Preview
  23. Q23The probability distribution function of a discrete random Variable $X$ is : $f(x)=\begin{cases}2k, & x=1\\3k, & x=3\\4k, & x=5\\0, & \text{…Preview
  24. Q24Suppose the probability mass function of the discrete random variable is : | $X=x$ | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | $p(x)…Preview
  25. Q25The value of $E(6x)$ is : (a) $0$ (b) $6\,E(x)$ (c) $36\,E(x)$ (d) $E(x)$Preview
  26. Q26The probability function of a random variable is defined as : | $X = x$ | $-1$ | $-2$ | $0$ | $1$ | $2$ | | --- | --- | --- | --- | --- | --…Preview
  27. Q27The discrete random variable X has the probability function | $X$ | $1$ | $2$ | $3$ | $4$ | | --- | --- | --- | --- | --- | | $P(X = x)$ | $…Preview
  28. Q28A coin is tossed thrice. Let X be the number of observed heads. Find the cumulative distribution function of X.Preview

More questions

+Show 7 questions7 questions
  1. Example 1Distinguish between a discrete random variable and a continuous random variable. Give one business-related example of each.Free
  2. Example 2Two fair coins are tossed together, and $X$ denotes the number of heads obtained. Construct the probability distribution (probability mass f…Free
  3. Example 3For the probability distribution of $X$ (the number of heads in two coin tosses) obtained in the previous example, find the mathematical exp…Free
  4. Example 4For the same distribution of $X$ (number of heads in two coin tosses), find the variance $Var(X)$ and the standard deviation $SD(X)$.Preview
  5. Example 5A discrete random variable $X$ has the probability distribution shown below. Construct its cumulative distribution function (c.d.f.), and us…Preview
  6. Example 6A continuous random variable $X$ has probability density function $f(x) = 2x$ for $0 \leq x \leq 1$, and $f(x)=0$ elsewhere. (i) Verify that…Preview
  7. Example 7A shopkeeper offers a simple game as part of a promotional lottery: a customer pays ₹10 to roll a single fair die once. If the die shows a 6…Preview