Consider the following random variables: U approximately (5, 10) V approximately

Question

Consider the following random variables: U approximately (5, 10) V approximately Exponential(5) W approximately Binomial(10, 0.2) (a) What is the mean and variance of each? (b) Consider now, X_n = sigma^n_i=1 Xi, where X is either U, V or W and different X_i are assumed indepdnent. What is the mean and variance of this random sum (a function of n)? For X either U, V or W. define, X^approximately_n = X_n - E(X_n)/squareroot var(X_n) Use the CLT to postulate the distribution of X^approximately_n for non-small n. Generate Monte Carlo estimates of P(|X^approximately_n|> 2.0) using no less than 106 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

Answer:

Let U is Uniform(5,10).

Mean(U) = (5+10)/2 = 15/2 = 7.5

Variance(U) = (10-5)^2/12 = 25/12 = 2.08

Let V is Exponential(5)

Mean(V) = 1/5 = 0.2

Variance(V) = 1/5^2 = 1/25 = 0.04

Let W is Binomial (10,0.2)

Mean(W) = 10*0.2 = 2

Variance(W) = 10*0.2*0.8 = 1.6

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