6.2 Linear combinations of random variables · Combine two small distributions with sliders a and b, build aX + bY from every outcome pair, and see why variances add and why 2X is not X₁ + X₂.
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About this lab
Combine two small distributions with sliders a and b, build aX + bY from every outcome pair, and see why variances add and why 2X is not X₁ + X₂.
You will be able to
Use E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X), and explain why b shifts the distribution without changing the variance and why a negative a never gives a negative variance.
Use E(aX + bY) = aE(X) + bE(Y) and, for independent X and Y, Var(aX + bY) = a²Var(X) + b²Var(Y), checking both against the mean and variance of the distribution built from all outcome pairs.
Explain why Var(2X) = 4Var(X) differs from Var(X₁ + X₂) = 2Var(X), and why the variance formula fails when Y depends on X.
State that independent normal variables combine to a normal variable with mean aμ₁ + bμ₂ and variance a²σ₁² + b²σ₂², and that independent Poisson variables add, X + Y ~ Po(m₁ + m₂), while a combination such as 2X + Y is not Poisson.
