Let X,Y,Z be three independent (i.e. mutually independent) random variables, each uniformly distributed on the interval [0,1].

Question

1. Find the mean and variance of 1/(Z+1).
E[1/(Z+1)]=
var(1/(Z+1))=
2. Find the mean of XY/(Z+1).
Hint: Use your answer to the previous part, together with the independence assumption.
E[XY/(Z+1)]=
3. Find the probability that XY/Z≤1. Enter a numerical answer.
P(XY/Z≤1)=

Answer 1

1.
E[1/(Z+1)] = 1/2
var(1/(Z+1)) = 0.0195
2.
E[XY/(Z+1)] = 0.1733
3.
P(XY/Z≤1) = 0.75

Answer 2

1. ln(2)
2. 0.5+ln(2)^2
3. 0.5^2*ln(2)

Answer 3

Anyone care to share the rest of the answers to the test here?