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

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.

Answer:

Step 1 of 3:

(a)

In this question, we are asked to find the probability density function for given cumulative distribution function of logistic distribution.

Where may be any real number; the parameter may be any positive number..

Let be a random variable with this distribution.

We know the formula for calculating the cumulative distribution function (CDF) is:

Then probability density function can be found as a =

=

= ) assume t = ]

= ] (using chain rule of differentiation)

= []

= []

= ……………(1)

Hence probability distribution function =