TU Wien:Econometrics for Business Informatics VU (Schneider)/Exercise 1 (2023S)

1.1

Var(X) = 𝔼(X²) - 𝔼(X)²

Var(X) = 𝔼((X - 𝔼(X))²)

Var(X) = 𝔼(X² - 2𝔼(X)X + 𝔼(X)²)

Var(X) = 𝔼(X²) + 𝔼(-2𝔼(X)X) + 𝔼(𝔼(X)²)

Var(X) = 𝔼(X²) + (-2) * 𝔼(X) * 𝔼(X) + 𝔼(X)² * 𝔼(1)

Var(X) = 𝔼(X²) - 2𝔼(X)² + 𝔼(X)²

Var(X) = 𝔼(X²) - 𝔼(X)²

Be aware that you also need to know all definitions and why you are allowed to do certain steps. She'll ask.

1.2

Bernoulli experiment: There can only be two outcomes. Example: Flipping a coin (1 = number, 0 = head) or the chance of rolling a dice and gettig a 6.

X = 0 | 1

P (X = 1) = p

𝔼(X) = p

Var(X) = 𝔼(X²) - 𝔼(X)² ---> 𝔼(X²) = P(X = 1) = p (Since X = 0 or 1, therefore X²=X)

= p - p²

= p * (1 - p)