1. Using the axioms of probability, prove P(A ∪ B) = P(A) + P(B) P(A ∩ B).

2. In a binary communication channel, 0s and 1s are transmitted with equal probability. The probability that a 0 is correctly received (as a 0) is 0.75. The probability that a 1 is correctly received (as a 1) is 0.99. Suppose we receive a 1, what is the probability that, in fact, a 0 was sent?

3. A continuous random variable X has probability density function

f(x) = α (1 - x),  x ∈ [0, 1/2]

Determine:

(a) The value of α;

(b) P(1/3 < X < 1/2).

4. Using the inverse-transform method on uniform (0, 1) (pseudo)random variates, simulate 1000 outcomes from X distributed as in 3, and estimate the probability in 3(b) by computing the sample mean of the indicator If_{{1/3 < x < 1/2}}. This is the crude Monte Carlo estimator of the probability. Give a typical estimate, and supply your code.

5. By simulating 10000 sent-and-received bits in the binary communications channel of 2, estimate the probability in 2 via crude Monte Carlo. Give a typical estimate, and supply your code.

6. Consider the following model for the arrival of calls at a telephone exchange. During each of the 1-second time slots [0, 1), [1, 2), [2, 3), ....., either 1 call arrives (with probability 0.005) or no calls arrive. Assume that the number of calls in the time slots are independent of each other.

(a) Calculate the probability that we have to wait more than 3 minutes before the first call arrives.

(b) Denote by Y the number of seconds we have to wait until we receive the first call. Calculate Var(Y ) exactly and estimate this quantity based on 100000 simulations of this process via crude Monte Carlo. Make sure to supply a typical estimate as well as your code.