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.