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Mathematics and Statistics · Class 12 Commerce

Ch 16Probability Distributions — Class 12 Mathematics and Statistics, concept-first.

In the earlier study of probability we assigned probabilities to events — subsets of a sample space. Very often, however, what we really care about is a number attached to each outcome: the number of heads when three coins are tossed, the number of defective bulbs in a carton, the daily demand for a product, the lifeti…

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Key concepts

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Random Variables and Probability Distributions

A random variable turns the outcomes of a chance experiment into numbers, so that questions like 'how likely are 6 heads in 8 tosses' can be answered with a formula instead of a listing. This chapter builds discrete rand…

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CA Foundation

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Chapter contents

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

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Random Variables — Discrete and Continuous

In the earlier study of probability we assigned probabilities to events — subsets of a sample space. Very often, however, what we really care about is a number attached to each outcome: the number of…

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Probability Distribution of a Discrete Random Variable (p.m.f.)

For a discrete random variable taking the values (or countably many values), the probability mass function (p.m.f.) states the probability of each value:

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Probability Density Function of a Continuous Random Variable (p.d.f.)

A continuous random variable takes uncountably many values, so we can no longer attach a positive probability to each single value — in fact for a continuous , for any particular number .

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Expected Value (Mean) and Variance

A distribution is summarised by two numbers: the expected value (mean), which locates its centre, and the variance, which measures its spread.

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Binomial Distribution

Many experiments consist of repeating the same trial a fixed number of times, where each trial has only two possible results — conventionally "success" and "failure".

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Poisson Distribution

The Poisson distribution models the number of times a rare event happens over a fixed interval of time, length, area or volume — the number of accidents at a junction per week, misprints per page, tel…

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Introduction to the Normal Distribution

The normal distribution is the most important continuous distribution in statistics. Heights, weights, measurement errors, examination marks and many natural and economic quantities are found to be ap…

Exercises

+Show 5 questions5 questions
  1. Q10A random variable $X$ has probability mass function $P(X=x)=\dfrac{x}{15}$ for $x=1,2,3,4,5$. (i) Verify it is a valid p.m.f. (ii) Find $P(X…Free
  2. Q11A die is thrown $4$ times. Getting a \"six\" is considered a success. Find the probability of getting (i) exactly $2$ sixes, (ii) at most $1…Free
  3. Q12The number of telephone calls received at an exchange follows a Poisson distribution with an average of $3$ calls per minute. Find the proba…Preview
  4. Q13The weekly wages of workers are normally distributed with mean $\overline{X}=\text{Rs. }1000$ and standard deviation $\sigma=\text{Rs. }200$…Preview
  5. Q14A continuous random variable $X$ has p.d.f. $f(x)=3x^2$ for $0\le x\le 1$ and $f(x)=0$ otherwise. (i) Verify it is a valid p.d.f. (ii) Find…Preview

Sample & Board Papers

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

+Show 23 questions23 questions
  1. Q1The following function represents the p.d.f of a.r.v. X $f(x) = \begin{cases} kx, & \text{for } 0 < x < 2 \\ 0, & \text{otherwise} \end{case…Preview
  2. Q2If $X \sim B\left(20, \dfrac{1}{10}\right)$, then $E(X) =$ ______ (a) 2 (b) 5 (c) 4 (d) 3Preview
  3. Q3State whether the following statement is true or false: If $X \sim P(m)$ with $P(X = 1) = P(X = 2)$ then $m = 1$. (a) True (b) FalsePreview
  4. Q4The probability distribution of a discrete r.v. X is as follows: | x | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | |…Preview
  5. Q5Solve the following problem: An examination consists of 5 multiple choice questions, in each of which the candidate has to decide which one…Preview
  6. Q6A random variable X has the following probability distribution: | $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | --- | --- | --- | --- | --- | --- | --…Preview
  7. Q7Solve the following problem : If X follows Poisson distribution such that $P(X = 1) = 0.4$ and $P(X = 2) = 0.2$, find variance of X.Preview
  8. Q8A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of getting at least two success. Soluti…Preview
  9. Q9A random variable $X$ has the following probability distribution: | $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | --- | --- | --- | --- | --- | --- |…Preview
  10. Q10The eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at least one d…Preview
  11. Q11If $X \sim P(m)$ with $P(X = 1) = P(X = 2)$, then find the mean and $P(X = 2)$. Given $e^{-2} = 0.1353$ Solution: Since $P(X = 1) = P(X = 2)…Preview
  12. Q12The expected value of the sum of two numbers obtained when two fair dice are rolled is ______. (a) 5 (b) 6 (c) 7 (d) 8Preview
  13. Q13Find the mean of the number of heads in three tosses of a fair coin.Preview
  14. Q14Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that all the five cards ar…Preview
  15. Q15Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has: no defect…Preview
  16. Q16The probability distribution of a discrete r.v. X is as follows: | x | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | |…Preview
  17. Q17The following is the p.d.f. of a r.v. X. $f(x) = \begin{cases} \frac{x}{8}, & \text{for } 0 < x < 4 \\ 0, & \text{otherwise.} \end{cases}$ F…Preview
  18. Q18The following is the p.d.f. of a r.v. X. $f(x) = \begin{cases} \frac{x}{8}, & \text{for } 0 < x < 4 \\ 0, & \text{otherwise.} \end{cases}$ F…Preview
  19. Q19The following is the p.d.f. of a r.v. X. $f(x) = \begin{cases} \frac{x}{8}, & \text{for } 0 < x < 4 \\ 0, & \text{otherwise.} \end{cases}$ F…Preview
  20. Q20In a town, 10 accidents take place in the span of 50 days. Assuming that the number of accidents follows Poisson distribution, find the prob…Preview
  21. Q21Determine the value of $k$ for the following probability distribution of $X$: | $X = x$ | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | ---…Preview
  22. Q22A fair coin is tossed 5 times. Find the probability of obtaining exactly 5 heads.Preview
  23. Q23A random variable X follows Poisson distribution such that – $P[X = 2] = \left(\dfrac{3}{4}\right) \cdot P[X = 1]$. Find $P[X > 1]$. [Use: $…Preview

More questions

+Show 9 questions9 questions
  1. Example 1Three coins are tossed simultaneously. Let $X$ denote the number of heads. Construct the probability distribution of $X$ and verify that it…Free
  2. Example 2A discrete random variable $X$ has probability distribution $P(X=x)=cx$ for $x=1,2,3,4$, and $P(X=x)=0$ otherwise. Find (i) the constant $c$…Free
  3. Example 3The probability distribution of a random variable $X$ is given below. Find its mean $E(X)$, variance and standard deviation.\n\n| $X=x$ | $1…Free
  4. Example 4A continuous random variable $X$ has probability density function $f(x)=kx$ for $0\le x\le 3$ and $f(x)=0$ otherwise. Find (i) the value of…Preview
  5. Example 5For a binomial distribution the mean is $4$ and the variance is $2$. Find $n$, $p$ and $q$, and hence write $P(X=0)$.Preview
  6. Example 6A fair coin is tossed $6$ times. Find the probability of getting (i) exactly $2$ heads, (ii) at least $1$ head.Preview
  7. Example 7The average number of accidents at a certain junction is $2$ per week. Assuming a Poisson distribution, find the probability that in a given…Preview
  8. Example 8$2\%$ of the items produced by a machine are defective. In a random sample of $100$ items, use the Poisson approximation to find the probabi…Preview
  9. Example 9The marks of students in a test are normally distributed with mean $\mu=50$ and standard deviation $\sigma=10$. Find the probability that a…Preview