TN 12th Maths — Learn, Visualize, Practice

1. Random variables

In an experiment whose outcome is uncertain, we often care about a number attached to each outcome — the number of heads in five tosses, the lifetime of a bulb, the sum on two dice. A rule that assigns a real number to every outcome of a sample space is called a random variable.

DefinitionRandom variable

A random variable $X$ is a function $X : S \to \mathbb{R}$ that assigns a real number $X(s)$ to each outcome $s$ in the sample space $S$ .

Random variables come in two flavours. A discrete random variable takes isolated (countable) values such as $0, 1, 2, \dots$ . A continuous random variable can take any value in an interval of the real line.

Discrete vs continuous

Discrete: number of defective items, number of heads, score on a die — you can list the values.
Continuous: height, weight, time, temperature — values fill an interval and cannot be listed one by one.

Key idea: A random variable turns messy outcomes into plain numbers, so we can average them, spread them, and compute probabilities with algebra and calculus.

2. Discrete distributions and the pmf

For a discrete random variable we describe its behaviour by listing each value together with its probability. This list is the probability distribution, and the function giving those probabilities is the probability mass function (pmf).

DefinitionProbability mass function

For a discrete random variable $X$ , the pmf is $p(x) = P(X = x)$ .

A valid pmf must satisfy two conditions:

Conditions for a pmf

p(x) \ge 0 \quad \text{and} \quad \sum_{x} p(x) = 1

Note: Every probability is between 0 and 1, and because some value must occur, all the probabilities add up to exactly 1. Use this to find a missing probability.

ExampleVerify a pmf and find a missing value

X

takes values

0, 1, 2, 3

with

p(x) = 0.1, 0.3, k, 0.2

. Find

k

and confirm it is a valid pmf.

3. Continuous distributions and the pdf

For a continuous random variable, the probability of hitting any single exact value is 0 — there are infinitely many possibilities. Instead we use a probability density function $f(x)$ and measure probability as area under its curve.

DefinitionProbability density function

A function $f(x)$ is a pdf for a continuous random variable $X$ if $f(x) \ge 0$ for all $x$ and the total area under it is $1$ .

Conditions for a pdf

f(x) \ge 0 \quad \text{and} \quad \int_{-\infty}^{\infty} f(x)\,dx = 1

Probability over an interval is the area under the curve on that interval:

P(a \le X \le b) = \int_{a}^{b} f(x)\,dx

Key idea: Discrete probabilities are heights you add up; continuous probabilities are areas you integrate. In both cases the grand total is 1.

ExampleFind the constant in a pdf

f(x) = kx^2

on

0 \le x \le 3

and

0

elsewhere. Find

k

.

4. Cumulative distribution function

Sometimes we want the probability of being at or below a value. The cumulative distribution function (cdf) accumulates probability from the left.

DefinitionCumulative distribution function

The cdf is $F(x) = P(X \le x)$ .

How to compute F(x)

Discrete: $F(x) = \sum_{t \le x} p(t)$ — add the probabilities of all values up to $x$ .
Continuous: $F(x) = \int_{-\infty}^{x} f(t)\,dt$ — the area to the left of $x$ .
$F$ is non-decreasing, with $F(-\infty) = 0$ and $F(\infty) = 1$ .
For continuous $X$ , $f(x) = F'(x)$ — the density is the slope of the cdf.

Note: $P(a < X \le b) = F(b) - F(a)$, so the cdf makes interval probabilities a simple subtraction.

5. Expectation, variance and standard deviation

The expectation (mean) $E(X)$ is the long-run average value of the random variable — a probability-weighted balance point. The variance measures how widely the values spread around that mean.

Expectation

E(X) = \sum_{x} x\,p(x) \quad \text{(discrete)}, \qquad E(X) = \int x\,f(x)\,dx \quad \text{(continuous)}

To measure spread, we first find $E(X^2)$ (weight the squared values), then use:

Variance and standard deviation

\text{Var}(X) = E(X^2) - [E(X)]^2, \qquad \sigma = \sqrt{\text{Var}(X)}

Watch out: $E(X^2)$ is not the same as $[E(X)]^2$. Square each value first, weight by probability, then subtract the square of the mean.

ExampleMean and variance of a small table

X: 1, 2, 3

with

p(x): 0.2, 0.5, 0.3

. Find

E(X)

,

E(X^2)

,

\text{Var}(X)

and

\sigma

.

6. The binomial distribution

Many experiments are a fixed number of independent yes/no trials — tossing a coin $n$ times, testing $n$ items for defects. If each trial has the same success probability $p$ , the number of successes follows a binomial distribution.

Conditions for a binomial experiment

A fixed number $n$ of trials.
Each trial has just two outcomes: success or failure.
The success probability $p$ is the same on every trial.
The trials are independent of one another.

DefinitionBinomial probability

If $X$ is the number of successes, then for $x = 0, 1, \dots, n$ :

P(X = x) = \binom{n}{x} p^{x} (1-p)^{\,n-x}

Here $\binom{n}{x}$ counts the ways to place $x$ successes among $n$ trials, $p^x$ is the chance those succeed, and $(1-p)^{n-x}$ the chance the rest fail. The mean and variance have neat closed forms:

Mean and variance of a binomial

E(X) = np, \qquad \text{Var}(X) = np(1-p), \qquad \sigma = \sqrt{np(1-p)}

ExampleExactly 2 heads in 5 tosses

A fair coin is tossed

5

times. Find the probability of exactly

2

heads.

ExampleMean and variance of a binomial

A die is rolled

18

times; a success is rolling a six (

p = \tfrac16

). Find the mean and variance of the number of sixes.

7. Other distributions (a note)

The binomial is one of a family of standard models. When trials are rare but numerous (large $n$ , small $p$ ), the Poisson distribution approximates the binomial. When measurements cluster symmetrically around a mean, the continuous normal distribution (the bell curve) is the workhorse of statistics. In Class 12 the binomial is the central discrete example, so master its formula, mean $np$ , and variance $np(1-p)$ .

Key idea: Recognise the model first — is it a fixed count of yes/no trials? If so, reach for the binomial and its

np,\ np(1-p)

shortcuts.