Class 12 Maths and Statistics Part 2 Chapter 8: Probability Distributions

Key concepts: Probability Distributions

Random variable

A real-valued function defined on the sample space of a random experiment, X : S → R. It is discrete if it takes countable values and continuous if it takes all values in an interval.

Probability mass function

For a discrete r.v., p(xᵢ) = P(X = xᵢ) with p(xᵢ) ≥ 0 and Σp(xᵢ) = 1.

Cumulative distribution function

F(x) = P(X ≤ x) = Σ p(xᵢ) over xᵢ ≤ x. F is non-decreasing.

Expected value and variance

E(X) = μ = Σxᵢpᵢ. Var(X) = Σ(xᵢ − μ)²pᵢ = Σxᵢ²pᵢ − μ² = E(X²) − [E(X)]². S.D. = √Var(X).

Continuous random variable

Described by a probability density function f(x) ≥ 0 with ∫f(x) dx = 1 over its range; P(a < X < b) = ∫[a, b] f(x) dx and F(x) = ∫ f(t) dt up to x.

Binomial distribution

For n independent Bernoulli trials with success probability p (q = 1 − p): P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ, x = 0, 1, ..., n. Mean = np, variance = npq. Written X ~ B(n, p).

Poisson distribution

P(X = x) = e⁻ᵐ mˣ/x!, x = 0, 1, 2, ... with m > 0. Mean = variance = m. It approximates the binomial when n is large and p is small, with m = np.

Quick revision: important points

• For a binomial distribution the variance is less than the mean (npq < np).
• P(X = 0) for Poisson = e⁻ᵐ.
• Σp = 1 is used to find an unknown constant in a p.m.f.; ∫f(x) dx = 1 for a p.d.f.
• Bernoulli trials: two outcomes, constant p, independent trials.
• If mean = np and variance = npq are given, q = variance/mean.

Practice MCQs with answers

1. A random variable is a:

   1. Set of outcomes
   2. Real-valued function on the sample space
   3. Probability
   4. Constant
   Show answer

   Answer: B. Real-valued function on the sample space
   X : S → R.

2. The number of heads obtained in 3 tosses of a coin is a:

   1. Continuous random variable
   2. Discrete random variable
   3. Constant
   4. Density function
   Show answer

   Answer: B. Discrete random variable
   It takes values 0, 1, 2, 3.

3. The height of a person is an example of a:

   1. Discrete random variable
   2. Continuous random variable
   3. Binomial variable
   4. Poisson variable
   Show answer

   Answer: B. Continuous random variable
   It can take any value in an interval.

4. For a probability mass function, ΣP(X = x) equals:

   1. 0
   2. 1
   3. n
   4. The mean
   Show answer

   Answer: B. 1
   Total probability is 1.

5. The expected value E(X) of a discrete random variable is:

   1. Σx
   2. Σx p(x)
   3. Σp(x)
   4. Σx² p(x)
   Show answer

   Answer: B. Σx p(x)
   Probability-weighted average.

All Class 12 Maths and Statistics chapters

• Part 1 Chapter 1: Mathematical Logic
• Part 1 Chapter 2: Matrices
• Part 1 Chapter 3: Differentiation
• Part 1 Chapter 4: Applications of Derivatives
• Part 1 Chapter 5: Integration
• Part 1 Chapter 6: Definite Integration
• Part 1 Chapter 7: Application of Definite Integration
• Part 1 Chapter 8: Differential Equation and Applications
• Part 2 Chapter 1: Commission, Brokerage and Discount
• Part 2 Chapter 2: Insurance and Annuity
• Part 2 Chapter 3: Linear Regression
• Part 2 Chapter 4: Time Series
• Part 2 Chapter 5: Index Numbers
• Part 2 Chapter 6: Linear Programming
• Part 2 Chapter 7: Assignment Problem and Sequencing
• Part 2 Chapter 8: Probability Distributions

Class 12 Maths and Statistics notes: all chapters