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Welcome to your Linear Combinations of Random Variables (Senior Class Maths Study by Topics)

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#### Linear Combinations of Random Variables

###### 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/n10²/250 = 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