A distribution that has been used to model tolerance levels in bioassays is the logistic
Chapter 4, Problem 23(choose chapter or problem)
A distribution that has been used to model tolerance levels in bioassays is the logistic distribution with parameters \(\alpha \text { and } \beta\). The cumulative distribution function of the logistic distribution is
\(F(x)=\left[1+e^{-(x-\alpha) / \beta}\right]^{-1}\)
The parameter \(\alpha\) may be any real number; the parameter \(\beta\) may be any positive number. Let X be a random variable with this distribution.
a. Find the probability density function \(f X(x)\).
b. Show that \(f X(x)\) is symmetric around \(\alpha\), that is, \(f X(\alpha-x)=f X(\alpha+x)\) for all x.
c. Explain why the symmetry described in part (b) shows that \(\mu X=\alpha\). You may assume that \(\mu X\) exists.
Equation Transcription:
Text Transcription:
\alpha and \beta
F(x)=[1+e^{-(x-\alpha) / \beta}t]^{-1}
\alpha
\beta
f X(x)
f X(x)
\alpha
f X(\alpha-x)=f X(\alpha+x)
\mu X=\alpha
\mu X
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