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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 pi such that pi≥0 and Σpi=1.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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 pi such that pi≥0 and Σpi=1.

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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 pi such that pi≥0 and Σpi=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.

Rule to remember

For a discrete distribution, pi≥0 and Σpi=1. Mean: E(X)=Σxipi. Second moment: E(X^2)=Σxi^2pi. Variance: Var(X)=E(X^2)-[E(X)]^2.

Memory hook

Distribution first, mean next, square table before variance.

Examples and method

Worked example

Let X take values 1, 2, 3 with probabilities 1/6, 1/2, 1/3. Check validity: 1/6+1/2+1/3=1/6+3/6+2/6=1. Mean: E(X)=1(1/6)+2(1/2)+3(1/3)=1/6+1+1=13/6. E(X^2)=1^2(1/6)+2^2(1/2)+3^2(1/3)=1/6+2+3=31/6. Variance = 31/6-(13/6)^2 = 186/36-169/36=17/36.

Method to apply

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

Diagram support

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.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

Make a table with columns xi, pi, xipi, and xi^2pi. This prevents missing terms in E(X) and Var(X).

Quick check

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

E(X)=0×1/4+1×1/2+2×1/4=0+1/2+1/2=1.

Answer writing and exam use

1-mark answer

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 pi such that pi≥0 and Σpi=1.

2-mark answer

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 pi such that pi≥0 and Σpi=1. For a discrete distribution, pi≥0 and Σpi=1. Mean: E(X)=Σxipi. Second moment: E(X^2)=Σxi^2pi. Variance: Var(X)=E(X^2)-[E(X)]^2. 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-mark answer

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. For a discrete distribution, pi≥0 and Σpi=1. Mean: E(X)=Σxipi. Second moment: E(X^2)=Σxi^2pi. Variance: Var(X)=E(X^2)-[E(X)]^2. Let X take values 1, 2, 3 with probabilities 1/6, 1/2, 1/3. Check validity: 1/6+1/2+1/3=1/6+3/6+2/6=1. Mean: E(X)=1(1/6)+2(1/2)+3(1/3)=1/6+1+1=13/6. E(X^2)=1^2(1/6)+2^2(1/2)+3^2(1/3)=1/6+2+3=31/6. Variance = 31/6-(13/6)^2 = 186/36-169/36=17/36. Questions may ask to construct a probability distribution, find an unknown probability using Σpi=1, calculate mean or variance, or interpret X from a given experiment. Calculating variance as Σ(xi-E(X)) without multiplying by probabilities, or using E(X^2)-E(X) instead of E(X^2)-[E(X)]^2.
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