Given that a random variable have a Poisson distribution with ;

(i) Find [co()] if () = ln 2

(ii) Suppose is a random variable with probability density function given by; 1 −1− , ≥ 0 (x) = {Γ() . . . 0 , oℎ Where is a fixed positive constant. Show that; Γ( + ) 1 + ( ) , = 0,1, 2 (x) = {Γ()Γ( + 1) 2 . . . 0 , oℎ

Hence find the distribution of (x), if is an integer or ∀ ∈ ℤ

has a discrete Uniform distribution on the integers 0, 1, 2, 3, … , and has a discrete Uniform distribution on 1, 2, 3, … , .

Show that ( ) − ( ) = (2n+1)/12

Consider the number of successes in x + m Bernoulli trials each with a success probability . Suppose that is a random variable with uniform distribution over the interval (0, 1). Find and identify P( = x).

Suppose that the random variable W has a beta distribution with probability density function () given as () = K (1 − ) , 0 < < 1

Find the value of K and hence the average value of W. A 5 a) Let {; =]1, 2, 3, … } be a sequence of random variables.

Explain what is means to say that

(i) has a limiting distribution as → [infinity].

(ii) The limiting distribution of is degenerate. b) The random variable has probability density function −(x−theta) , x > 0 (x) = { . . . 0 , oℎ Using the moment generating functions, show that

(i) has a limiting distribution which is degenerate at x = theta

(ii) = ( − theta) has a limiting distribution which is exponential with mean 1.

c) Given that ; = 1, 2, 3, … are independent and identically distributed random variables with () = and () = < [infinity], show that the distribution of , ∑=1 − = √ Converges to the standard normal distribution function as → [infinity]. Hint: Let 1, 2, 3, … be a sequence of random variables having moment generating function m(), m2(), m3(), … respectively. If lim m() = m() →[infinity] then the distribution function of converges to the distribution function of

Suppose that ; = 1, 2, 3, … are independent random variables with a common uniform distribution over (0, 1) and is distributed over (0, −1) with 0 = 1 , = 1, 2, … , . Let = ∏ , = 1, 2 , 3, … =1 Show by mathematical induction induction or any other method that has probability density function (− z)−1 (z) = ( − 1)! , = 1, 2 , 3, … Hence deduce that and have the same distribution ( = 1, 2 , 3, … )

Suppose that customers arrive at a bank at a Poisson rate . Assume that two customers arrived during the first ten minutes. What is the probability that,

(i) Both customers arrived during the first two and half (2 ) minutes.

(ii) At least one customer arrived during the first five minutes.

A random variable theta~ (− , ), find the probability density function of the random variable 2 2 = tan theta

An insurance company pays out claims on its life insurance policies in accordance with a Poisson process having rate = 2 per week. If the amount of money paid on each policy is uniformly distributed between GHS 1,000.00 and GHS 10,000.00, find the mean amount of money paid by the insurance company in a fine-week period.

Let (Ω, , P) be a probability space and H ∈ with P(H) > 0. For any arbitrary subset, ∈ , and define P( ∩ H) PH() = P(|H) = P(H) Then show that (H, H, PH) is a probability space.

A 11 Suppose a fair coin is flipped twice. Let 1 = "Head on 1st Toss" 2 = "Head on 2nd Toss" 3 = "Exactly one Head" Show that 1, 2, 3are Pairwise independent but not Mutually independent