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Mean and variance of a random variable: Class 12 practice

For a discrete random variable, mean E(X) = Σ x·P(x) and variance Var(X) = Σ x²P(x) − [E(X)]². Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Fractions

The idea behind mean and variance of a random variable

For a discrete random variable, mean E(X) = Σ x·P(x) and variance Var(X) = Σ x²P(x) − [E(X)]². The probabilities must add to 1.

Step-by-step method

  1. Check ΣP = 1.
  2. E(X) = Σ xP(x).
  3. Var = Σ x²P(x) − E(X)².

Worked example 1 (Basic)

X takes values 0, 1, 2 with probabilities 2/5, 2/5, 1/5. Find E(X).

  1. E(X) = 4/5
  2. Answer: 4/5

Worked example 2 (Standard)

X takes values 0, 1, 2, 3 with probabilities 1/4, 1/5, 1/4, 3/10. Find E(X) and Var(X).

  1. E(X) = 8/5
  2. Var = 1.34
  3. Answer: E(X) = 8/5, Var = 1.34

Worked example 3 (Challenge)

X takes values 0, 1, 2, 3 with probabilities 5/12, 5/12, 1/12, 1/12. Find E(X) and Var(X).

  1. E(X) = 5/6
  2. Var = 0.81
  3. Answer: E(X) = 5/6, Var = 0.81

Common mistakes and misconceptions

  • Forgetting to subtract E(X)²
  • Probabilities not summing to 1

Practice questions

  1. (Basic) X takes values 0, 1, 2 with probabilities 1/2, 3/10, 1/5. Find E(X).
  2. (Basic) X takes values 0, 1, 2 with probabilities 4/11, 5/11, 2/11. Find E(X).
  3. (Standard) X takes values 0, 1, 2, 3 with probabilities 2/7, 3/14, 3/7, 1/14. Find E(X) and Var(X).
  4. (Standard) X takes values 0, 1, 2, 3 with probabilities 5/13, 6/13, 1/13, 1/13. Find E(X) and Var(X).
  5. (Challenge) X takes values 0, 1, 2, 3 with probabilities 3/19, 6/19, 4/19, 6/19. Find E(X) and Var(X).
  6. (Challenge) X takes values 0, 1, 2, 3 with probabilities 1/3, 5/18, 1/6, 2/9. Find E(X) and Var(X).

Answer key with full working

  • Q1: E(X) = 7/10 → Answer: 7/10
  • Q2: E(X) = 9/11 → Answer: 9/11
  • Q3: E(X) = 9/7 → Var = 0.92 → Answer: E(X) = 9/7, Var = 0.92
  • Q4: E(X) = 11/13 → Var = 0.75 → Answer: E(X) = 11/13, Var = 0.75
  • Q5: E(X) = 32/19 → Var = 1.16 → Answer: E(X) = 32/19, Var = 1.16
  • Q6: E(X) = 23/18 → Var = 1.31 → Answer: E(X) = 23/18, Var = 1.31

Syllabus note

This skill is listed under "Probability" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is mean and variance of a random variable in?

"Probability", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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