1. a) (5 marks) Prove the following total probability rule for calculating the probability P(A), based on P(A|E) and P(A|Ec ):

P(A) = P(A|E)P(E)+ P(A|Ec )P(Ec ).

b) (10 marks) Extend the total probability rule above for calculating the conditional probability P(A|F), based on P(A|E \ F) and P(A|Ec \ F). Prove this new rule.

c) (10 marks) Among the people who su↵er from diabetes, 55% are female and 45% are male. 10% of the females with diabetes su↵er from heart disease. 20% of the males with diabetes su↵er from heart disease. What is the probability that a randomly selected person su↵ers from heart disease, given that this person has diabetes? (Hint: Use b).)

2.  a) (5 marks) Give the definition of the density function of a continuous random variable X.

b) (15 marks) Let X be a continuous random variable with density function fX (x). Derive the density function of the variable Y = X4 .

c) (10 marks) Let X be a standard normal random variable, i.e. X is a continuous random variable with density function:

fX (x) = e−x2 /2 ,  −1 < x < 1.

Find E(Y), where Y = X4 .  (Hint:  Use b) and the fact that R01 x1/2e−x dx = T(3/2) =

^⇡/2.)

function

fX (x) = {0(2)x

if 0 < x < 1

otherwise

Find P ⇣X(4) − X(1)  < a⌘ ,    (Hint: find first F(x) and then fX(1), X(4)(x,y)).

8  1

fX,Y (x,y) = <  T

( 0

if x2 + y2  < 1

otherwise

Find the joint density function of the polar coordinates R =  (X2  + Y2 )1/2   and ✓ =

6.  (15 marks) a) Given the density function