Discrete random variables

Question 1

What is coding?

Answer 1

Taking a set of data amd peforming poerations on it to change it

Question 2

What happens to E(X) when it is transformed by E(2X) and how do you prove it?

Hint: use identity for E(X)

Answer 2

E(X) = Σx / n

E(2X) = Σ2x / n = 2Σx / n

Therefore E(2X) = 2E(X)

Question 3

What happens when E(X) is transform by E(X-1)

Hint: use identity for E(X)

Answer 3

E(X) = Σx / n

E(X-1) = Σ(x-1) / n = Σ(x) - n / n

Therefore E(X-1) = E(X) - 1

Question 4

What happens to Var(X) under the transformation Var(2X+1)

Answer 4

Var(2X+1)=4Var(X)

Var(X) is scaled by the square of the square of the coefficient, however shifting the “position” of the data set makes no difference to how it varies

Proof here: https://math.stackexchange.com/questions/1708266/why-square-a-constant-when-determining-variance-of-a-random-variable

Question 5

E(X₁ +- X₂) = ?

Answer 5

E(X₁ +- X₂) = E(X₁) +- E(X₂)

Question 6

Var(X₁ +- X₂) = ?

Answer 6

Var(X₁) + Var(X₂)

Sign doesn’t matter variances are always summed