Study Notes
Linear combinations of random variables involve operations on random variables to produce new random variables. These combinations can be used to calculate expectations and variances.
- Expectation of a linear combination — The expected value of a linear combination of random variables is the sum of the expected values multiplied by their coefficients. Example: E(aX + bY) = aE(X) + bE(Y)
- Variance of a linear combination — The variance of a linear combination of independent random variables is the sum of the variances multiplied by the square of their coefficients. Example: Var(aX + bY) = a²Var(X) + b²Var(Y)
- Normal distribution property — If X is normally distributed, then aX + b is also normally distributed. Example: If X ~ N(μ, σ²), then aX + b ~ N(aμ + b, a²σ²)
- Poisson distribution property — The sum of two independent Poisson random variables is also Poisson distributed. Example: If X ~ Po(λ) and Y ~ Po(μ), then X + Y ~ Po(λ + μ)
Exam Tips
Key Definitions to Remember
- Expectation of a linear combination: E(aX + bY) = aE(X) + bE(Y)
- Variance of a linear combination: Var(aX + bY) = a²Var(X) + b²Var(Y)
- Normal distribution property: aX + b is normally distributed if X is
- Poisson distribution property: Sum of independent Poisson variables is Poisson
Common Confusions
- Forgetting to square the coefficients when calculating variance
- Assuming non-independent variables can use the same variance formula
Typical Exam Questions
- What is E(2X + 3Y) if E(X) = 5 and E(Y) = 4? Answer: 2(5) + 3(4) = 22
- How do you find Var(3X - 2Y) if Var(X) = 9 and Var(Y) = 4? Answer: 3²(9) + (-2)²(4) = 81 + 16 = 97
- If X ~ N(10, 4), what is the distribution of 2X + 3? Answer: N(23, 16)
What Examiners Usually Test
- Ability to apply formulas for expectation and variance
- Understanding of properties of normal and Poisson distributions
- Solving problems involving linear combinations of random variables