Solved – CDF(expected value) = 0.5

Question

For a continuous random variable x with CDF F(x), the expected value is x0, can I say F(x0) = 0.5?

If not, does the value x1 that makes CDF(x1) = 0.5 mean anything special?

Answer 1

does the value x1 that makes CDF(x1) = 0.5 mean anything special?

Any value for which it's true is a median; for distributions that aren't continuous (such as discrete distributions) there may be no value at which you get exact equality. Consider the uppermost face on a toss of a fair six-sided die with faces numbered 1,2,3,1,2,3 $-$ the cdf of that variable is never exactly $\frac12$. More generally if there's no value at which it is $\frac12$ you find the first value for which the cdf exceeds $\frac12$.

See wikipedia on the median

For a continuous random variable x with CDF F(x), the expected value is x0, can I say F(x0) = 0.5?

Not in general. It's sometimes true that the expected value and the median are the same but typically it's not the case.

A sufficient condition is that the distribution is symmetric and the mean exists. Symmetry is not a necessary condition though.