Solved: Let X have a logistic distribution with pdf Show

Chapter 5, Problem 6E

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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.

Equation Transcription:

 

 

Text Transcription:

X  

f(x)=e^-x/(1+e^-x)2^, - infinity <x< infinity

Y=1/1+e^-X  

U(0,1)

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.

Equation Transcription:

 

 

Text Transcription:

X  

f(x)=e^-x/(1+e^-x)2^, - infinity <x< infinity

Y=1/1+e^-X  

U(0,1)

ANSWER:

Step 1 of 3

Define a variable Y for the logistically distributed variable X over   as,

Thus, Y is defined over .

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