Question

5. You are given that X1 and X2 follow an exponential distribution with mean 10 and 20, respectively. Both Tommy and Tony want to create a new distribution using X1 and X2 to model the waiting time (in mins), X, at a doctor's office. (i) Based on Tommy's Judgement: 80% of the chance, the waiting time follows a distribution as X1 and 20% of the chance, it follows the same distribution as X. (ii) Tony would like to use a splicing method: the density of X is proportional to the density of Xi in (0, 15) and proportional to the density of X2 in (15, oo). (iii) The probability that X is no more than 15 is the same under both methods. Using Tony's method, calculate the probability that X is larger than 20

Answer 1

By Tony's methode we have

$P(X>20)=P(X_2>20)\\ ~~~~~~~~~~~~~~~~~~~ =\int_{20}^{\infty}{0.05}e^{-0.05x_2} dx_2\\ ~~~~~~~~~~~~~~~~~~~ =e^{-0.05\times 20}(=e^{-1})\\ ~~~~~~~~~~~~~~~~~~~\approx 0.3679$