1. Use the Monte Carlo method to estimate the following: Jo exp(exp(x)) dx this easy version of the Monte Carlo method, uses the law of large numbers: Law of Large Numbers: if {X; } are i.i.d. UNiform on [0, 1], n Σh(X;) → i=1 ["₁ The more complete Monte Carlo method allows to compute h(x)dx = E(h(X)) ["h h(x)f(x)dx = E(h(X)) for other random variebles, not only Uniform. Monte Carlo uses a Markov chain to generate sapling from the "steady state" distribution. See slides attached. In this problem, you are only asked to try the Law of Large Numbers.

Let X = 1 if a randomly selected vehicle passes an emissions test and X = 0 otherwise. Then X is a Bernoulli rv with pmf p(1) = p and p(0) = 1 - p. (a) Compute E(X²). E(X²) = (b) Show that V(X) = p(1 − p). V(X) = E = = : p(1 - p) (c) Compute E(X98). E(X98) = - [E(X)]²

T(n) = 2T(n − 1) + 1, T(0) = 1. Use the iteration method to find a () evaluation for T(n).

A chemical supply company currently has in stock 100 Ib of a certain chemical, which it sells to customers in 4-lb batches. Let X = the number of batches ordered by a randomly chosen customer, and suppose that X has the following pmf. 1 2. 3 4 P(x) 0.3 0.5 0.1 0.1 Compute E(X) and V(X). E(X) = V(X) = batches batches? Compute the expected number of pounds left after the next customer's order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of X.] expected weight left Ib wariance of weight left Ib2