Mathematics · Class 12 Science
Ch 10Random Variables and Probability Distributions — Class 12 Mathematics, concept-first.
A great many real questions boil down to attaching a number to the outcome of an experiment whose result cannot be predicted with certainty: how many heads turn up in 20 coin tosses, how many defective items are in a batch of 50, how many customers arrive at a shop in an hour.
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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Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
A great many real questions boil down to attaching a number to the outcome of an experiment whose result cannot be predicted with certainty: how many heads turn up in 20 coin tosses, how many defectiv…
What is a Random Variable?
A great many real questions boil down to attaching a number to the outcome of an experiment whose result we cannot predict with certainty.
Probability Distribution of a Discrete Random Variable
A discrete random variable is one whose range is either a finite list of numbers or a list that can be written out one after another (countably infinite), such as .
Mean and Variance of a Discrete Probability Distribution
Once a probability distribution is written down, two summary numbers describe it well: where it is centred, and how spread out it is.
Bernoulli Trials
Many experiments of everyday and scientific interest have only two possible results: a coin shows heads or tails, a manufactured item is defective or not, a patient responds to a treatment or does not…
The Binomial Distribution
Suppose independent Bernoulli trials are performed, each with probability of success and of failure. Let be the random variable counting the total number of successes in the trials; can take any of th…
Mean and Variance of the Binomial Distribution
For a general discrete distribution, computing the mean and variance means summing and term by term, as in section 10.3.
The Poisson Distribution
The Binomial distribution needs a precisely known, finite number of trials , together with the probability of success on each one.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1A random variable $X$ has the probability distribution $X:\ 0,\,1,\,2$; $P(X):\ k,\,2k,\,3k$. Find $k$.Preview
- Q2Find the mean of the binomial distribution $B\left(n=6,\ p=\dfrac{1}{3}\right)$.Preview
- Q3If a random variable $X$ follows a Poisson distribution with mean $\lambda=4$, find $P(X=0)$.Preview
- Q4Find the mean and variance of the number of heads in 3 tosses of a fair coin.Preview
- Q5A random variable $X$ has the probability distribution $X:\ 0,\,1,\,2,\,3,\,4$; $P(X):\ 0.1,\,k,\,2k,\,2k,\,k$. Find $k$ and the mean of $X$…Preview
- Q6A fair die is thrown 3 times. If getting "a number greater than 4" is called a success, find the probability distribution of the number of s…Preview
- Q7If $X\sim B(n,p)$ is a binomial random variable, prove that its mean is $np$.Preview
- Q8A Poisson variable satisfies $P(x = 1) = P(x = 2)$. Find $P(x = 5)$.Preview
- Q9A random variable $x$ has the following probability distribution : $X = x$: $0, 1, 2, 3, 4, 5, 6, 7$ with $P(X = x)$: $0,\ k,\ 2k,\ 2k,\ 3k,…Preview