Solved: Let X have a logistic distribution with pdf f(x) = ex (1 + ex)2 , < x < . Show

Chapter 5, Problem 5.1-6

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QUESTION:

Let X have a logistic distribution with pdf

               \(f(x) = \frac {e^{-x}}{(1 + e−x)^2}\),\(\quad -\infty < x < \infty\).

Show that

                    \(Y = \frac {1}{1 + e^{-X}}\)

has a U(0, 1) distribution.

Questions & Answers

QUESTION:

Let X have a logistic distribution with pdf

               \(f(x) = \frac {e^{-x}}{(1 + e−x)^2}\),\(\quad -\infty < x < \infty\).

Show that

                    \(Y = \frac {1}{1 + e^{-X}}\)

has a U(0, 1) distribution.

ANSWER:

Step 1 of 3

Let X be the random variable with PDF,defined over

Now, define,

For the defined range of X, Y lies between 0 and 1.

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