Question

Let X₁, X₂ be two independent normal random variables with means μ₁, μ₂ and standard deviations σ₁, σ₂, respectively. Consider Y = X₁ – X₂; μ₁ = μ₂ =1, σ₁ = 1, σ₂ = 2. Then,

1. Y is normally distributed with mean 0 and variance 1
2. Y is normally distributed with mean 0 and variance 5
3. Y has mean 0 and variance 5, but is NOT normally distributed
4. Y has mean 0 and variance 1, but is NOT normally distributed

Answer 1

Correct Answer - Option 2 : Y is normally distributed with mean 0 and variance 5

Concept:

Var(X) = E(X²) − E(X)²

Var (X + Y) = Var(X) + Var(Y) + 2Cov (X, Y)

Cov (X, Y) = E (XY) − E(X)E(Y)

If random variables X and Y are independent then cov (X, Y) = 0.

For independent random variables X and Y, the variance of their sum or difference is the sum of their variances i.e. Var (X ± Y) = Var(X) ± Var(Y)

Calculation:

For the given questions

X₁ and X₂ independent normal variables

Y = X₁ – X₂

For subtractive operation in normal variables the nature remains the same i.e. Normal.

So, Y is also normally distributed.

Given, μ₁ = μ₂ = 1; σ₁ = 1; σ₂ = 2

Y = X₁ – X₂

Mean (Y) = Mean (X₁) – Mean (X₂)

⇒ μ(Y) = μ(X₁) – μ(X₂)

μ(Y) = 1 – 1 = 0

Similarly, Var (Y) = Var (X₁) + Var (X₂) = σ₁² + σ₂²

Var (Y) = 1² + (-2)² = 5

Where σ = Standard deviation, Variance = σ²