X1, X2 and X3 are three independent random variables. Xi ~ N (4, 12), for i=1, 2, 3.?

Since X₁, X₂, X₃ are independent normal random variables with μ = 4 and σ² = 12 for each random variable, Y = (X₁ + X₂ + X₃)/3 is also a normal random variable, with

μ_Y = (1/3)(4 + 4 + 4) = 4, and

(σ_Y)² = (1/3)² (12 + 12 + 12) = 4.

(Remember that if a and b are constants and X, Y are random variables, then

μ_(aX + bY) = a μ_X + b μ_Y, and [σ_(aX + bY)]² = a² (σ_X)² + b² (σ_Y)²;

this extends to finitely many random variables.)

That is, E(Y) = μ_Y = 4, and StdDev(Y) = σ_Y = 2.

I hope this helps!