medium difficulty7 min

Random Variable and Probability Distribution

A random variable is a real-valued function on the outcomes of a random experiment. For a discrete random variable X, its probability distribution lists values xi with probabilities π such that π≥0 and Σπ=1.

In one line: Distribution first, mean next, square table before variance.

Practice This Concept

Understand it

A random variable assigns numbers to outcomes so that numerical measures can be calculated. The mean E(X) gives the long-run average value, while variance Var(X) measures spread around the mean. A valid probability distribution must include all possible values and probabilities adding to 1.

Example

If X is the number of heads in two coin tosses, then X can be 0, 1, or 2. The probabilities are P(X=0)=1/4, P(X=1)=1/2, and P(X=2)=1/4.

Formula and rule

For a discrete distribution, π≥0 and Σπ=1.

Mean: E(X)=Σxipi.

Second moment: E(X²)=Σxi²π.

Variance: Var(X)=E(X²)-[E(X)]².

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let X take values 1, 2, 3 with probabilities 1/6, 1/2, 1/3.

1Check validity: 1/6+1/2+1/3=1/6+3/6+2/6=1.
2Mean: E(X)=1(1/6)+2(1/2)+3(1/3)=1/6+1+1=13/6.
3E(X²)=1²(1/6)+2²(1/2)+3²(1/3)=1/6+2+3=31/6.
4Variance = 31/6-(13/6)² = 186/36-169/36=17/36.

Method you can reuse

1List all possible values of X.
2Assign probability to each value.
3Check all probabilities are non-negative and sum to 1.
4Compute xipi for the mean.
5Compute xi²π for the second moment.
6Use Var(X)=E(X²)-[E(X)]² and simplify.

Visual tip

A table is more useful than a diagram. Bar graphs may show probability distribution shape, with values of X on the horizontal axis and probabilities on the vertical axis.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For X taking values 0, 1, 2 with probabilities 1/4, 1/2, 1/4, find E(X).

Think first, then show answer

E(X)=0×1/4+1×1/2+2×1/4=0+1/2+1/2=1.

Common wrong answer

Calculating variance as Σ(xi-E(X)) without multiplying by probabilities, or using E(X²)-E(X) instead of E(X²)-[E(X)]².

Don't mix this up

Students sometimes write outcomes instead of values of X, or forget to check that the probabilities add to 1 before finding mean and variance.

Exam tip

Make a table with columns xi, π, xipi, and xi²π. This prevents missing terms in E(X) and Var(X).

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask to construct a probability distribution, find an unknown probability using Σπ=1, calculate mean or variance, or interpret X from a given experiment.

1
Enough for 1 mark
A random variable is a real-valued function on the outcomes of a random experiment. For a discrete random variable X, its probability distribution lists values xi with probabilities π such that π≥0 and Σπ=1.
2
Add for 2 marks
Example: If X is the number of heads in two coin tosses, then X can be 0, 1, or 2. The probabilities are P(X=0)=1/4, P(X=1)=1/2, and P(X=2)=1/4.
3
Add for 3 marks
A random variable assigns numbers to outcomes so that numerical measures can be calculated.

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