Some definitions and results for your reference.
If X ~ N(a, σ^2) where a ∈ R and σ > 0 then
- pdf (probability density function) of X is
f(x) = (1/√(2π)σ) * e^(-(x-a)^2 / (2σ^2))
∀ x ∈ R
- mgf (moment generating function) of X is M_X(t) = E(e^(tX)) = e^(at + (σ^2t^2)/2)
If X ~ Bernoulli(p), p ∈ [0,1], p + q = 1. Then the pmf of X is f_X(x) = C(p, q) * p^x * q^(1-x)
True or False? In the following questions, determine whether the statement is true or false, Give an example if the statement is true; otherwise, give a counterexample.
a) Let X1, X2 are independent. If E(Xi), Var(Xi) < ∞, i=1,2. Then Var(X1-2X2-3) = Var(X1) + 4Var(X2) - 3.
b) If X, Y are independent then E(XY) = E(X)E(Y) if all these expectations exist and E(Y) ≠0.
c) Let Z ~ N(0,1) and F is the cdf of Z. Then |F(x)| = 1 for any x ∈ R.
