8 - combining random variables

Question 1

adding random variables

Answer 1

E(aX + bY + c) = aE(X) + bE(Y) + c
Var(aX + bY + c) = a²Var(X) + b²Var(Y)

Question 2

adding observations of a random variable

Answer 2

E(X1+X2) = E(X1)+E(X2)
Var(X1+X2) = Var(X1)+Var(X2) not Var(2X)

Question 3

sample mean

Answer 3

E(sample mean) = E(X)
Var(sample mean) = Var(X)/n

Question 4

unbiased estimate of the pop variance

Answer 4

s² = n/n-1 σ²

Question 5

linear combinations of normal variables

Answer 5

if X and Y are random variables and follow a normal distribution
if Z = aX + bY + c then Z is also normally distributed

Question 6

central limit theorem

Answer 6

for any distribution if n>25 then the sample mean can be said to be normally distributed with mean μ and variance σ²/n