Consider the following random variables: U ~ Uniform(5, 10) V ~ Exponential(5) W

Question

Consider the following random variables: U ~ Uniform(5, 10) V ~ Exponential(5) W ~ Binomial(10, 0.2) (a) What is the mean and variance of each? (b) Consider now, X_n = sigma_i = 1^n X_i, where X is either U, V or W and different X_i are assumed indepdnent. What is the mean and variance of tills random sum (a function of n)? (c) For X either U, V or W, define, X_n = X_n - E(X_n)/Squareroot var(X_n) Use the CLT to postulate the distribution of X_n for non-small n. (d) Generate Monte Carlo estimates of P[|X_n| > 2.0) using no less than 10^6 generations of X_n for every n, (separately for each U, V or W). Compare your results to P{|Z| > 2.0) taken from a normal distribution table, where Z is a standard normal random variable. Do this for n = 5, 10, 20. Tabulate and explain your results.

Explanation / Answer

Solution

Part (a)

Back-up Theory

If X ~ Uni(a, b). E(X) = {(a + b)/2}; V(X) = {(a - b)2/12} ………………………………..(1)

If X ~ B(n, p). then

Mean (average) of X = E(X) = np…………………………………………………………..(2)

Variance of X = V(X) = np(1 – p)…………………………………………………………..(3)

If X ~ Exponential with parameter , E(X) = V(X) = ……………………………………(4)

So, [vide (1) under Back-up Theory], Mean U = 7.5 and V(U) = 2.0833 ANSWER 1

      [vide (4) under Back-up Theory], Mean V = 5 and V(V) = 5 ANSWER 2

      [vide (2) and (3) under Back-up Theory], Mean W = 2 and V(W) = 1.6 ANSWER 3

Part (b)

Back-up Theory

E{[1,n](X)} = [1,n]{E(X)} = nE(X) and V{[1,n](X)} = [1,n]{V(X)} = nV(X) …… (5)

So, [vide (1) under Back-up Theory],

Mean Xn and V(Xn) are respectively,

7.5n and 2.0833n if X = U;

5n and 5n if X = V and

2n and 1.6n if X = W ANSWER

Part (c)

By Central Limit Theorem, for sufficiently large n, {X – E(x)}/SD(X) ~ N(0, 1) ANSWER

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