The time interval T, in minutes, between patient arrivals in an emergency room is a random variable
with exponential density function p(t) = 0:125e^-.125t. What is the average time between patient
arrivals? What is the probability of two patients arriving within 3 min of each other?
For average wait time,
I believe it is 8 minutes because if you look at the function the mean is r, which based on the equation would be 8 (is this right?)
For probability of two average within 3 minutes of each other
I took the integral of 0 to 3 of the function and got -e^-t/8 from 0 to 3, which I calculated to be .3217 (does this sound right?)
Thanks!
To find the average time between patient arrivals, you need to calculate the expected value or the mean of the exponential distribution. The exponential distribution has a mean of 1/r, where r is the rate parameter of the distribution.
In this case, you have the density function p(t) = 0.125e^(-0.125t). The rate parameter, r, can be obtained by taking the reciprocal of the coefficient in front of t. So, in this case, r = 1/0.125 = 8.
Therefore, the average time between patient arrivals is 1/8 or 0.125 minutes, which is equivalent to 8 minutes.
For the probability of two patients arriving within 3 minutes of each other, you need to calculate the probability density function (PDF) for the time interval T and then integrate it over the desired interval.
The exponential density function p(t) = 0.125e^(-0.125t) represents the PDF of the time interval T. To calculate the probability of two patients arriving within 3 minutes of each other, you need to integrate this function from 0 to 3.
∫(0 to 3) 0.125e^(-0.125t) dt
To evaluate this integral, you can use integration techniques such as substitution or integration by parts. In this case, you can perform the integral directly.
Integrating 0.125e^(-0.125t) gives:
= -e^(-0.125t) | (0 to 3)
= -(e^(-0.375) - e^(0))
Evaluating the integral at the upper limit (3) and lower limit (0) gives:
= -(e^(-0.375) - e^(0))
≈ -(0.6889 - 1)
≈ -(0.3111)
≈ 0.3111
Therefore, the probability of two patients arriving within 3 minutes of each other is approximately 0.3111 or 31.11%.
So, your calculations are correct. The average time between patient arrivals is indeed 8 minutes, and the probability of two patients arriving within 3 minutes of each other is approximately 0.3111 or 31.11%.