Question

TRUE or FALSE? Do not explain your answer.

(a) If A and B are any independent events, then P(A ∪ B) = P(A) + P(B).

(b) Every probability density function is a continuous function.

(c) Let X ∼ N(0, 1) and Y follow exponential distribution with parameter λ = 1. If X and Y are independent, then the m.g.f. MXY (t) = e t 2 /2 1 1−t .

(d) If X and Y have moment generating functions MX and MY , respectively, then M′ X+Y (0) = M′ X(0)+ M′ Y (0).

(e) The following function F is the cumulative distribution function of some random variable: F(x) = ⎧⎪⎪ ⎨ ⎪⎪⎩ e x−1 , if 0 < x < 1, 0, otherwise.

(f) If Cov(X, Y ) = 0, then X and Y are independent.

Answer 1

a)If A and B are independent,

False

b)TRUE, the pdf is for a continuous variable and is a continuous function. Probability mass function is for a discrete variable and is a discrete function.

d)FALSE

e)F(0)=e^-1 and F(1)=1. The probabilty needs to start from 0 and reach 1. Hence, this is not a cdf. False

f)If Cov(X,Y)=0, the events are mutually exclusive but need not be independent. False