Let X1X2X100 be the outcome of 100 independent tosses of a f

Let X₁,X₂,...,X₁₀₀ be the outcome of 100 independent tosses of a fair coin. Pr(Xi = 0) =Pr(X_i = 1) = 0.5. Define X = X₁ - X₂.

E(X) =

var(X) =

E : Expected Value, var: Variance

Probide explanation.

Solution

E[X1] = P(X1=0) * 0 + P(X1=1) * 1 = 0.5*0 + 0.5*1 = 0.5

E[X2] = P(X2=0) * 0 + P(X2=1) * 1 = 0.5*0 + 0.5*1 = 0.5

E[X} = E[X1 - X2] = E[X1] - E[X2] = 0.5 - 0.5 = 0

E[X1²] = P(X1=0) * 0² + P(X1=1) * 1² = 0.5*0 + 0.5*1 = 0.5

Var(X1) = E(X1²) - [E(X1)]² = 0.5 - 0.5² = 0.25

E[X2²] = P(X2=0) * 0² + P(X2=1) * 1² = 0.5*0 + 0.5*1 = 0.5

Var(X2) = E(X2²) - [E(X2)]² = 0.5 - 0.5² = 0.25

Var(X) = Var(X1 - X2) = Var(X1) + Var(X2) - 2Cov(X1 , X2)

Cov(X1 , X2) = 0 as X1 and X2 are independent.

Var(X) = 0.25 + 0.25 = 0.5