Expectation and Variance of a Function of X

• E(aX + b) = aE(X) + b
• Var(aX + b) = a²Var(X)

Examples

1. The random variable T has a mean of 5 and a variance of 16. Find the two pairs of values for the constants c and d such that E(cT + d) = 100 and Var(cT + d) = 144
   • Expand the expectation equation:
     • E(cT + d) = cE(T) + d = 100
     • ∴ 5c + d = 100
   • Expand the variance equation:
     • Var(cT + d) = c²Var(T) = 144
     • 16c² = 144
     • ∴ c = ±3
   • Use the first equation to find the two pairs:
     • when c = +3, d = 85
     • when c = -3, d = 115

Combinations of Random Variables

• Expectations of combinations of random variables:
  • E(aX + bY) = aE(X) +  bE(Y)
• Variance of combinations of independent random variables:
  • Var(aX + bY + c) = a²Var(X) + b²Var(Y)
  • Var(X ± Y) = Var(X) + Var(Y)
• Combinations of identically distributed random variables having mean μ and variance σ²:
  • E(2X) = 2μ and E(X₁) + E(X₂) = 2μ
  • Var(2X) = 4σ² but Var(X₁ + X₂) = 2σ²

Examples

1. It is given that X₁ and X₂ are independent, and E(X₁) = E(X₂) = μ, Var(X₁) = Var(X₂) = σ². Find E(X) and Var(X), where X = ½(X₁ + X₂)
   • Split the variance and expectation into individual components:
     • Var(½(X₁ + X₂)) = (½)²Var(X₁) + (½)²Var(X₂)
     • E(½(X₁ + X₂)) = ½E(X₁) + ½E(X₂)
   • Substitute the given values:
     • Var(½(X₁ + X₂)) = ¼σ² + ¼σ² = σ²
     • E(½(X₁ + X₂)) = ½μ + ½μ = μ

Expectation and Variance of Sample Mean

• E(x̄) = μ
• Var(x̄) = ^σ²/_n

Examples

1. The mean weight of a soldier may be taken to be 90 kg, and σ = 10 kg. 250 soldiers are on board an aircraft, find the expectation and variance of their weight. Hence, find μ and σ of the total weight of soldiers
   • Let X be the average weight, therefore:
     • E(x̄) = μ = 90
     • Var(x̄) = ^σ²/_n = ^10²/₂₅₀ = 0.4 kg²
   • To find the μ of the total weight:
     • E(X₁) + E(X₂) + … + E(X₂₅₀) = 250E(X) = 22500 kg
   • To find σ, find Var(X) first:
     • Var(X₁) + … + Var(X₂₅₀) = 250Var(X) = 2500 kg
     • Var(X) = σ² = 25000
     • ∴ σ = √25000 = 158.1 kg