__Expectation and Variance of a Function of X__

__Expectation and Variance of a Function of X__

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

__Examples__

__Examples__

- 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

- Expand the expectation equation:

__Combinations of Random Variables__

__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(X
_{1}) + E(X_{2}) = 2μ - Var(2X) = 4σ² but Var(X
_{1}+ X_{2}) = 2σ²

- E(2X) = 2μ and E(X

__Examples__

__Examples__

- It is given that X
_{1}and X_{2}are independent, and E(X_{1}) = E(X_{2}) = μ, Var(X_{1}) = Var(X_{2}) = σ². Find E(X) and Var(X), where X = ½(X_{1}+ X_{2})- Split the variance and expectation into individual components:
- Var(½(X
_{1}+ X_{2})) = (½)²Var(X_{1}) + (½)²Var(X_{2}) - E(½(X
_{1}+ X_{2})) = ½E(X_{1}) + ½E(X_{2})

- Var(½(X
- Substitute the given values:
- Var(½(X
_{1}+ X_{2})) = ¼σ² + ¼σ² = σ² - E(½(X
_{1}+ X_{2})) = ½μ + ½μ = μ

- Var(½(X

- Split the variance and expectation into individual components:

__Expectation and Variance of Sample Mean__

__Expectation and Variance of Sample Mean__

- E(x̄) = μ
- Var(x̄) =
^{σ²}/_{n}

__Examples__

__Examples__

- 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(X
_{1}) + E(X_{2}) + … + E(X_{250}) = 250E(X) = 22500 kg

- E(X
- To find σ, find Var(X) first:
- Var(X
_{1}) + … + Var(X_{250}) = 250Var(X) = 2500 kg - Var(X) = σ² = 25000
- ∴ σ = √25000 = 158.1 kg

- Var(X

- Let X be the average weight, therefore: