Statistics · Class 12 Commerce
Ch 6Random Variable and Discrete Probability Distribution — Class 12 Statistics, concept-first.
In many situations arising in business and economics, the outcome of a random experiment is more useful to us as a number than as a description in words. For example, when three items are picked at random for quality inspection, we are usually interested in how many of them are defective, not in which particular items…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Expectation and Variance of a Random Variable
The expectation (mean) of a discrete random variable is , the probability-weighted average of its values. Key property: .
Most relevant Q&A
- If $X$ is a random variable such that $E(X) = 5$ and $Var(X) = 4$, find $E(X^2)$. (A) 20 (B) 25 (C) 29 (D) 9Free
- If $X$ is a random variable such that $Y = 2X + 3$, and $E(X) = 4$, $Var(X) = 5$, find $E(Y)$ and $Var(Y)$.Preview
- A random variable $X$ has the following probability distribution: $X$: 1, 2, 3, 4; $P(X)$: 0.1, 0.3, 0.4, 0.2. Find $E(X)$ and $Var(X)$.Preview
- For the probability distribution of a discrete random variable $X$, $E(X) = 5$ and $E(X^2) = 35$. What will be the variance of this distribu…Preview
- There are 2 black and 2 white balls in a box. Two balls are drawn without replacement from it. Obtain probability distribution of the number…Preview
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 Variable
In many situations arising in business and economics, the outcome of a random experiment is more useful to us as a number than as a description in words.
Probability Mass Function
For a discrete random variable taking values , the probability mass function (pmf), written or , gives the probability with which takes each of its possible values.
Expectation (Mean) of a Random Variable
The expectation or mean of a discrete random variable , denoted or , is the probability-weighted average of the values can take.
Variance and Standard Deviation of a Random Variable
The mean alone does not tell us how spread out the values of are around it. The variance, denoted or , measures this spread:
Binomial Distribution
Many real situations consist of a fixed number of independent, identical trials, each of which ends in only one of two outcomes — usually called "success" and "failure" — with the same probability of…
Poisson Distribution
The Poisson distribution is used to model the number of occurrences of a relatively rare event in a fixed interval of time, length, area or volume — for example, the number of customers arriving at a…
Exercises
+−Show 6 questionsHide questions6 questions
- Q4If $X$ is a random variable such that $E(X) = 5$ and $Var(X) = 4$, find $E(X^2)$. (A) 20 (B) 25 (C) 29 (D) 9Free
- Q7In a large batch of items, 10% are known to be defective. If a random sample of 6 items is selected, find the probability that at most 1 ite…Free
- Q9A random variable $X$ follows a Poisson distribution with mean 4. Find $P(X=0)$ and $P(X=1)$. (Take $e^{-4} = 0.0183$.)Preview
- Q10Which of the following can be a valid probability distribution of a discrete random variable $X$ taking the values $0, 1, 2$? (A) $P(0)=0.5,…Preview
- Q11State the conditions that must be satisfied for a random experiment to follow a Binomial distribution.Preview
- Q12If $X$ is a random variable such that $Y = 2X + 3$, and $E(X) = 4$, $Var(X) = 5$, find $E(Y)$ and $Var(Y)$.Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 31 questionsHide questions31 questions
- Q1The binomial distribution has mean $5$ and variance $\frac{10}{7}$. What will be the type of this distribution? (a) Positively skewed (b) Ne…Preview
- Q2Define Bernoulli trials.Preview
- Q3The mean and variance of a binomial distribution are $3.9$ and $2.73$ respectively. Find the parameters of the distribution. OR Write any fo…Preview
- Q4Find the constant $K$ for the following discrete probability distribution. Hence obtain mean of this distribution: $P(x) = K \cdot {}^{4}P_{…Preview
- Q5The probability that the bomb dropped from a plane over a bridge will hit the bridge is $\frac{1}{5}$. Two bombs are enough to destroy the b…Preview
- Q6Which variable of the following will be an illustration of discrete variable? (a) Height of a student (b) Weight of a student (c) Blood pres…Preview
- Q7For the probability distribution of a discrete random variable $X$, $E(X) = 5$ and $E(X^2) = 35$. What will be the variance of this distribu…Preview
- Q8For a positively skewed Binomial Distribution with $n = 10$, which of the following values might be the value of mean? (a) 5 (b) 3 (c) 9 (d)…Preview
- Q9Define Bernoulli trials.Preview
- Q10The mean and variance of a binomial distribution are 3.9 and 2.73 respectively. Find the number of Bernoulli trials conducted in this distri…Preview
- Q11State properties of binomial distribution (Any six).Preview
- Q12The probability that a bomb dropped from a plane over a bridge will hit the bridge is $\dfrac{1}{5}$. Two bombs are enough to destroy the br…Preview
- Q13The binomial distribution has mean 5 and variance $\frac{10}{7}$. What will be the type of this distribution? (a) (A) Positively skewed (b)…Preview
- Q14Define discrete random variable.Preview
- Q15The probability distribution of a random variable $X$ is as follows: $p(x) = C(x^2 + x), \; x = -2, 1, 2$. Find the value of $C$. OR State t…Preview
- Q16There are 2 black and 2 white balls in a box. Two balls are drawn without replacement from it. Obtain probability distribution of the number…Preview
- Q17The parameter of binomial distribution of a random variable $X$ are $n = 4$ and $p = \frac{1}{3}$. Find the value of $P(X \le 2)$.Preview
- Q18Which variable of the following will be an illustration of continuous variable? (a) Number of accidents occurring at any place (b) Number of…Preview
- Q19For the probability distribution of a discrete random variable $X$, $E(X) = 5$ and $E(X^2) = 35$, what will be the variance of this distribu…Preview
- Q20The binomial distribution has mean 6 and variance $\frac{12}{7}$. What will be the type of this distribution? (a) Positively skewed (b) Nega…Preview
- Q21Mean of a symmetrical binomial distribution is 9. Find the value of its parameter $n$.Preview
- Q22State any four properties of binomial distribution.Preview
- Q23The mean and variance of the binomial distribution are 2 and $\frac{6}{5}$ respectively. Find $p(2)$ for this binomial distribution.Preview
- Q24A random variable $X$ denotes the number of accidents per year in a factory and the probability distribution of $X$ is given below : | $X =…Preview
- Q25Which variable of the following will be an illustration of discrete variable? (a) Height of a student (b) Weight of a student (c) Blood pres…Preview
- Q26Mean and variance of a discrete probability distribution are 3 and 7 respectively. What will be $E(X^2)$ for this distribution? (a) $10$ (b)…Preview
- Q27For which value of $x$, the value of $p(x)$ of binomial distribution with parameters $n = 4$ and $p = \dfrac{1}{2}$ becomes maximum? (a) $0$…Preview
- Q28The probability of failure in a binomial distribution is 0.6 and the number of trials in it is 5. Find the probability of success.Preview
- Q29Write any 2 properties of Bernoulli's Trials?Preview
- Q30The probability distribution of a random variable $X$ is defined as follows: $$p(x) = \dfrac{K}{(x+1)!},\ x = 1, 2, 3;\ K = \text{Constant}$…Preview
- Q31A person has kept 4 cars to run on rent. The probability that any car is rented during the day is 0.6. Find the probability that more-than o…Preview
More questions
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- Example 1Three fair coins are tossed simultaneously. If $X$ denotes the number of heads obtained, write the probability distribution of $X$.Free
- Example 2A discrete random variable $X$ has the probability distribution: $X = 0,1,2,3$ with $P(X) = k,\ 2k,\ 3k,\ 4k$ respectively. Find the value o…Free
- Example 3A random variable $X$ has the following probability distribution: $X$: 1, 2, 3, 4; $P(X)$: 0.1, 0.3, 0.4, 0.2. Find $E(X)$ and $Var(X)$.Preview
- Example 5A fair coin is tossed 5 times. Find the probability of getting exactly 3 heads.Preview
- Example 6For a binomial distribution with $n=10$ and $p=0.4$, find the mean and variance.Preview
- Example 8The average number of accidents at a certain road junction is 3 per week. Using the Poisson distribution, find the probability that in a giv…Preview