1. Tina has finally saved up enough money to make her dream of opening a music store come true. 'Tina's Trumpets' is incredibly popular! She measures the time t, in minutes, between customers arriving (i.e. the amount of time it takes for the next customer to arrive if a customer has just arrived), and tries to model it using a probability density function. She decides that the probability density function takes the form of either p(t) or r(t) where, for t > 0, p(t) = ae^(bt) and r(t) = m/(t + k)^2 , where a, b, m, and k are constants.

(a) How should p(t) and r(t) be defined for t < 0? Explain your answer.

(b) About 1/9 of the time, the next customer arrives within 10 seconds of the previous one. She uses this to approximate the value of her density functions at t = 0. Her approximation comes from the interpretation: "The probability of the next customer arriving between 0 and 10 seconds after the previous one is approximately 1/9 ." What should the value of her density functions at t = 0 be?

(c) Tina sets p(0) and r(0) equal to the value she found in (b), and solves for the values of a, b, m, and k. What values should she find? Remember that p(t) and r(t) are both probability density functions.

(d) Tina finds that the median amount of time between customers arriving is slightly over 1 minute. Based on this, which of p(t) and r(t) is a better model for the arrival time between customers?