hi I posted this question earlier:

what is the median waiting time for f(x)=((1/8)e^(-x/8))?

I said the answer was 5.55

is this correct? thank you!

and somebody helped me find the mean, but i was wondering if this was the right median.
thanks

To find the median waiting time for the given distribution function f(x) = (1/8)e^(-x/8), we need to calculate the cumulative distribution function (CDF) and then find the value of x at which the CDF is equal to 0.5.

Here's how you can do that:

Step 1: Calculate the cumulative distribution function (CDF)
The CDF is the integral of the probability distribution function (PDF). In this case, the PDF is f(x) = (1/8)e^(-x/8). To find the CDF, integrate f(x) from negative infinity to x.

CDF(x) = integral of f(t) dt from -infinity to x

CDF(x) = integral of ((1/8)e^(-t/8)) dt from -infinity to x

Integrating the PDF gives us:

CDF(x) = -e^(-t/8) from -infinity to x

CDF(x) = -e^(-x/8) - (-e^(-(-infinity)/8))

Since e^(-infinity) approaches zero, the second term becomes zero.

CDF(x) = -e^(-x/8)

Step 2: Find the median
The median is the value of x for which the CDF(x) equals 0.5. So, we can set the CDF(x) equal to 0.5 and solve for x.

0.5 = -e^(-x/8)

We can then solve for x by taking the natural logarithm of both sides:

ln(0.5) = -x/8

x = -8 * ln(0.5)

Using a calculator, we find:

x ≈ 5.541

Therefore, the median waiting time for the given distribution function f(x) = (1/8)e^(-x/8) is approximately 5.541, not 5.55.

I hope this clarifies things for you! Let me know if you have any further questions.