By: Tao Steven Zheng (郑涛)

【Problem】

Calculate the mean, median, and variance for the probability distribution function (PDF)

defined on the interval and is zero elsewhere.

【Solution】

(1) Mean

The mean is the first moment of the PDF.

(2) Median

To calculate the median, one must obtain the cumulative distribution function (CDF).

The median is located where the probability is 50%, so

(3) Variance

The variance is calculated by subtracting the square of the mean from the second moment.

Since ,