Experiment 3: Probability distributions
(1) Roll 12 dice simultaneously, and let X denotes the number of 6’s that appear. Calculate the probability of getting 7, 8 or 9, 6’s using R. (Try using the function pbinom; If we set S={get a 6 on one roll}, P(S)= 1/6 and the rolls constitute Bernoulli trials; thus X∼ binom(size=12, prob=1/6) and we are looking for P(7 ≤X≤ 9).
(3) On the average, five cars arrive at a particular car wash every hour. Let X count the number of cars that arrive from 10AM to 11AM, then X∼Poisson(λ = 5). What is probability that no car arrives during this time. Next, suppose the car wash above is in operation from 8AM to 6PM, and we let Y be the number of customers that appear in this period. Since this period covers a total of 10 hours, we get that Y ∼ Poisson(λ = 5×10 = 50). What is the probability that there are between 48 and 50 customers, inclusive?
(4) Suppose in a certain shipment of 250 Pentium processors there are 17 defective processors. A quality control consultant randomly collects 5 processors for inspection to determine whether or not they are defective. Let X denote the number of defectives in the sample. Find the probability of exactly 3 defectives in the sample, that is, find P(X = 3).
(a) How is X distributed?
(b) Sketch the probability mass function.
(c) Sketch the cumulative distribution function.
(d) Find mean, variance and standard deviation of X.
