91Ó°ÊÓ

Since the appointment duration follows an exponential distribution, the pdf can be written as: $$ f(x) = \frac{1}{\mu}e^{-\frac{x}{\mu}}, $$ where \(\mu\) is the mean of the distribution (30 minutes in this case) and \(x\) represents the appointment duration in minutes.

We are interested in finding the expected duration 1:30 appointment at the doctor's office. We can begin by finding the expected value of the second appointment duration, conditional on the duration of the first appointment: We integrate x multiplied by f2(x) times f1(y), and integrate from \(y=x\) to \(\infty\) dy then from 0 to x dx. $$ E[X_2|X_1] = \int_0^\infty x \left(\int_x^\infty \frac{1}{30}e^{-\frac{y}{30}} dy\right) \frac{1}{30}e^{-\frac{x}{30}} dx $$ Here, we have used two exponential random variables of mean 30 which are independent of each other. Now we need to compute the integral of \(E[X_2|X_1]\)

First, we compute the inner integral: $$ \int_x^\infty \frac{1}{30}e^{-\frac{y}{30}} dy = e^{-\frac{x}{30}} \\ $$ Now, substituting this back in our main expression: $$ E[X_2|X_1] = \int_0^\infty x \cdot e^{-\frac{x}{30}} \frac{1}{30}e^{-\frac{x}{30}} dx = \int_0^\infty x \cdot e^{-\frac{2x}{30}} \frac{1}{30} dx $$ We use integration by parts with u = x and \(dv = e^{-\frac{2x}{30}} \frac{1}{30}dx\). We now find du and v: $$ du = dx \\ v = \int e^{-\frac{2x}{30}} \frac{1}{30} dx = -\frac{15}{30} e^{-\frac{2x}{30}} = -\frac{1}{2} e^{-\frac{2x}{30}} \\ $$ Now, we compute the integral using integration by parts formula: $$ \int u dv = uv - \int v du $$ $$ E[X_2|X_1] = -\frac{1}{2} xe^{-\frac{2x}{30}}\Big|_0^\infty - \int_0^\infty (-\frac{1}{2}) e^{-\frac{2x}{30}} dx $$ The first term is zero. The second term is: $$ \int_0^\infty (-\frac{1}{2}) e^{-\frac{2x}{30}} dx = \frac{1}{2} \int_0^\infty e^{-\frac{2x}{30}} dx $$ Now we compute the last integral above: $$ \frac{1}{2} \int_0^\infty e^{-\frac{2x}{30}} dx = \left[-\frac{1}{2}\cdot \frac{30}{2} e^{-\frac{2x}{30}} \right]_0^\infty = 15 $$ This gives the expected duration for the second appointment, given the first appointment duration.

We have found that the expected duration for the second appointment, given the first appointment duration, is 15 minutes. Since both patients are on time, the 1:30 appointment has to wait for the first appointment to finish. Thus, the expected time the 1:30 appointment spends at the doctor's office is: (Expected time for the 1:30 appointment) = (30 min average duration of the first appointment) + (15 min expected duration for the second appointment given the first appointment duration) (Expected time for the 1:30 appointment) = 30 + 15 = 45 minutes Thus, the expected amount of time that the 1:30 appointment spends at the doctor's office is 45 minutes.