Suppose that customers arrive at a bank at an average rate of 4.3 per hour. Let X be a random variable that describes the number of customers arriving in the next hour, and let T be the random variable that describes the waiting time until the next customer arrives.

(a) What type of random variables are X and T , and what are the parameter(s)? Ans:

X is Poisson with = 4.3; T is Exponential with = 4.3

(b) Find the probability that 3 or less customers arrive in the next hour.

Ans: (3)=0.3771539

(c) Find the probability that the next customer arrives in less than a half an hour,

Ans: (<0.5)=0.8835158

Find the CDF of T. Ans: 0 <0 14.3 0

(d) Find the mean and median of T Ans: mean = 1/4.3 = 0.232558 hours, median = 0.161197.

(e) Find the 75th percentile of T, and describe what this number means. Ans: 0.322394, which means that 75% of the outcomes of T are less than 0.322394 hours.

(f) Find the probability that the next customer takes more than 20 minutes to arrive.

Ans: 0.238513

(g) Describe in words (>0.75|>0.25), then compute the value. Ans: The prob. that the next customer arrives in more than 0.75 hours from now if the customer does not arrive within 0.25 hours from now, which by the memoryless property is the prob. the next customer takes more than 0.5 hours is 4.30.5=0.11648.

(h) Suppose you keep track of the time you must wait for the next customer to arrive, and you do this for ten different customer arrivals. What is the probability that exactly 3 out of the 10 custom

Suppose pdf f(x) = 2 14. (f(x) is 0 outside of that domain.)

(a) Find the value of k that makes f(x) a valid PDF. Ans: 4/3

(b) Find the CDF. Ans: ()=0, <1;

F(x) = 4/3 (1 - 1/x ) for 14 ()=1 >4.

(c) If X is a random variable with f(x) as its PDF, compute:

(a) E(X). Ans: (4/3)ln(4)

(b) Var(X). Ans: 4 - [(4/3)ln(4)]2

(c) P(X=2). Ans: 0

(d) (2). Ans: 2/3

(e) (>2). Ans: 1/3

(f) (>2|3). Ans: 1/4