Please answer all four multiple choice questions below, questions 1a, 1b, 1c, and 1d. Please show all work and all steps.

1a.) Let X + Y be the sum of the above random variables X and Y. Find the mgf of X + Y.

a) Mx + My

b) e^2t

1b.) For a distribution function F, we have that F(x) geometrically represents:

a) the area under the graph of the density of f of X over the interval (-infinity, x]

b) the area under the graph of the density of f of X over the interval [x, +infinity)

1c.) Find f(ln 3) for the random variable X, which is defined as the density f(x) = 3e^-3x, for x>0, and 0 otherwise.

a) 0

b) e^-1

c) 1

1d.) Let X be an arbitray continuos random variable. Then the relation between P(X takes values in [3,5)) and P(X takes values in [3,5]) is the following:

a) the first one < the second one

b) they are equal

c) the first one > the second one