Cumulative Distribution Function of Logistic Distribution

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Theorem

Let $X$ be a continuous random variable with the logistic distribution.

Then the cumulative distribution function of $X$ is:

$\map {F_X} x = \dfrac 1 {1 + \map \exp {- \dfrac {x - \mu} s} }$

Proof

The derivative of $F_X$ is:

\(\ds \map {F'_X} x\) \(=\) \(\ds \paren {\frac 1 {1 + \map \exp {- \frac {x - \mu} s} } }'\)
\(\ds \) \(=\) \(\ds \frac {\paren 1' \paren {1 + \map \exp {- \frac {x - \mu} s} } - \paren 1 \paren {1 + \map \exp {- \frac {x - \mu} s} }'} {\paren {1 + \map \exp {- \frac {x - \mu} s} }^2}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {0 - \paren {\map \exp {-\frac {\paren {x - \mu} } s} }'} {\paren {1 + \map \exp {- \frac {x - \mu} s} }^2}\) Derivative of Constant, Sum Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {- \map {\exp'} {-\frac {\paren {x - \mu} } s} \paren {-\frac {\paren {x - \mu} } s}'} {\paren {1 + \map \exp {- \frac {x - \mu} s} }^2}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {\map \exp {- \frac {x - \mu} s} \paren {\frac x s - \frac \mu s}'} {\paren {1 + \map \exp {- \frac {x - \mu} s} }^2}\) Derivative of Exponential Function, Derivative of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\map \exp {- \frac {x - \mu} s} } {s\paren {1 + \map \exp {- \frac {x - \mu} s} }^2}\) Derivative of Constant Multiple, Derivative of Identity Function, Derivative of Constant
\(\ds \) \(=\) \(\ds \map {f_X} x\) Definition of Logistic Distribution

By the Fundamental Theorem of Calculus:

$\ds \int_a^b \map {f_X} x \rd x = \bigintlimits {\map {F_X} x} {x \mathop = a} {x \mathop = b}$

Therefore:

\(\ds \int_{-\infty}^x \map {f_X} \lambda \rd \lambda\) \(=\) \(\ds \lim_{a \to -\infty} \int_a^x \map {f_X} \lambda \rd \lambda\) Definition of Improper Integral
\(\ds \) \(=\) \(\ds \lim_{a \to -\infty} \paren {\map {F_X} x - \map {F_X} a}\) Result above
\(\ds \) \(=\) \(\ds \lim_{a \to -\infty} \map {F_X} x - \lim_{a \to -\infty} \map {F_X} a\) Sum Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds \map {F_X} x - \lim_{a \to -\infty} \frac 1 {1 + \map \exp {- \frac {a - \mu} s} }\) Limit of Constant Function
\(\ds \) \(=\) \(\ds \map {F_X} x - \lim_{b \to \infty} \frac 1 {1 + \map \exp b}\) Limit to Infinity of Linear Function, where $b = - \frac {a - \mu} s$
\(\ds \) \(=\) \(\ds \map {F_X} x - \lim_{c \to \infty} \frac 1 c\) Exponential Tends to Zero and Infinity, where $c = 1 + \map \exp b$
\(\ds \) \(=\) \(\ds \map {F_X} x\) Limit to Infinity of Reciprocal Function

Hence the result.

$\blacksquare$

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