A distribution that has been used to model tolerance levels in bioassays is the logistic distribution with parameters and . The cumulative distribution function

A distribution that has been used to model tolerance levels in bioassays is the logistic distribution with parameters α and β. The cumulative distribution function of the logistic distribution is
F(x) = [1 + e−(x−α)/β ]−1
The parameter α may be any real number; the parameter β may be any positive number. Let X be a random variable with this distribution.
a. Find the probability density function fX (x).
b. Show that fX (x) is symmetric around α, that is, fX(α − x) = fX(α + x) for all x.
c. Explain why the symmetry described in part (b) shows that μX = α. You may assume that μX exists.