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 =
