Solution Found!
Let X have a logistic distribution with pdf Show that has
Chapter 3, Problem 22E(choose chapter or problem)
Let X have a logistic distribution with pdf
\(f(x)=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}}\),\(\quad-\infty<x<\infty\).
Show that
\(Y=\frac{1}{1+e^{-X}}\)
has a U(0, 1) distribution.
Hint: Find \(G(y)=P(Y \leq y)=P\left(\frac{1}{1+e^{-X}} \leq y\right)\) when 0 < y < 1.
Questions & Answers
QUESTION:
Let X have a logistic distribution with pdf
\(f(x)=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}}\),\(\quad-\infty<x<\infty\).
Show that
\(Y=\frac{1}{1+e^{-X}}\)
has a U(0, 1) distribution.
Hint: Find \(G(y)=P(Y \leq y)=P\left(\frac{1}{1+e^{-X}} \leq y\right)\) when 0 < y < 1.
ANSWER:Answer
Given f(x)=, -
Here we have to show that y=
Step 1 of 2
P(Yy)=P(
y)
=P(1+e-x)
=P(e-x-1)
=P(x ln(
-1))