Logistic Distribution and Properties of Negative Exponential Distribution

Logistic Distribution

Logistic distribution is the limiting distribution of average of the smallest and the largest sample observation in a random sample size n. In other words, Logistic distribution is the limiting distribution of the standardized mid-range in a random sample of size n as n→∞.

Definition:

A continious random variable X having the probability density function (p.d.f.)

$$f(x) \ =\ \frac{ \\ {e^-}^{ \ \left ( \frac{x-\alpha}{\beta} \ \right )}}{ \ \beta \ \left [ \ 1 + {e^-}^{\left ( \ \frac{x-\alpha}{\beta} \ \right )} \ \right ]^2}\ \ ; \ -\infty \ < \ x \ < \ \infty, \ -\infty < \ \alpha \ < \infty \ and \ \beta \ > \ 0$$

is said to follow Logistic distribution with parametersα andβ.

The cummulative distribution function of the logistic variable X is given by

$$f(x) \ = \ \int_{-\infty}^{x} \ f(t) \ dt$$

$$= \ \int_{-\infty}^{x} \\ \frac{ \\ {e^-}^{ \ \left ( \frac{t-\alpha}{\beta} \ \right )}}{ \ \beta \ \left [ \ 1 + {e^-}^{\left ( \ \frac{t-\alpha}{\beta} \ \right )} \ \right ]^2} \ dt$$

$$= \ \int_{-\infty}^{x} \ \frac{{e^-}^{y}}{(1+{e^-}^{y})^2} \ dy \ \ \ where, \ y \ = \ \frac{t-\alpha}{\beta} \ \Rightarrow \ t \ = \ \alpha \ + \ \beta y \ \Rightarrow \ dt \ = \ \beta dy$$

$$putting \ z \ = \ 1 \ + \ {e^-}^{y}$$

$$\frac{dz}{dy} \ = \ {e^-}^{y} \ (-1) \ \Rightarrow \ -dz \ = \ {e^-}^{y} \ dy$$

$$= \ \int_{-\infty}^{x} \ - \ \frac{1}{z^2} \ dz$$

$$= \ \left [ \ \frac{1}{z} \ \right ]_{-\infty}^{x}$$

$$= \ \left [ \ \frac{1}{1 + {e^-}^{y}} \ \right ]_{-\infty}{x}$$

$$= \ \left [ \frac{1}{ \ 1 + {e^-}^{\left ( \ \frac{t-\alpha}{\beta} \ \right )}} \ \right ]_{-\infty}^{x}$$

$$= \ \frac{1}{ \ \ \ \ 1 + {e^-}^{\left ( \ \frac{x-\alpha}{\beta} \ \right )}} \ \ \ ; \ -\infty \ < \ x \ < \ \infty, \ -\infty < \ \alpha \ < \infty \ and \ \beta \ > \ 0$$

standard Logistic Distribution

If X has a Logistic distribution with probability density function

$$f(x) \ =\ \frac{ \\ {e^-}^{ \ \left ( \frac{x-\alpha}{\beta} \ \right )}}{ \ \beta \ \left [ \ 1 + {e^-}^{\left ( \ \frac{x-\alpha}{\beta} \ \right )} \ \right ]^2}$$

Let us define standard Logistic variable as

$$Z \ = \ \frac{x-\alpha}{\beta}$$

Then, the jocobion of transformation is

$$J \ = \ \frac{dx}{dz} \ = \ \beta$$

Now, the probability density function of Z is given by

$$f(z) \ = \ f(x) \ . \ |J|$$

$$= \ \frac{{e^-}^{z}}{\beta \ \left [ \ 1 + {e^-}^{z} \ \right ]^2} \ \beta$$

$$= \ \frac{{e^-}^{z}}{ \ \left [ \ 1 + {e^-}^{z} \ \right ]^2} \ \ \ \ ; \ -\infty \ < \ z \ < \ \infty$$

This is the probability density function of standard Logistic distribution.

Note:

Ifα = 0 andβ = 1 in Logistic distribution, then the resulting distribution is called standard Logistic distribution with pdf

$$f(x) \ = \ \frac{{e^-}^{x}}{ \ \left [ \ 1 + {e^-}^{x} \ \right ]^2} \ \ \ \ ; \ -\infty \ < \ x \ < \ \infty$$

$$Since, \ f(z) \ = \ \frac{{e^-}^{z}}{ \ \left [ \ 1 + {e^-}^{z} \ \right ]^2} \ =\ \frac{e^z}{ \ \left [ \ 1 + e^z \ \right ]^2} \ = \ f(-z)$$

the standard Logistic curve is symmetrical about the vertical axis through z = 0.

The distribution function of standard Logistic variable Z is

$$F(z) \ = \ \frac{1}{1 \ + \ {e^-}^{z}} \ \ \ \ ; \ -\infty \ < \ z \ < \ \infty$$

Properties of Logistic Distribution:

1. Mean of Logistic distribution isα but standard Logistic distribution has zero mean.
2. variance of Logistic distribution isβ²Π²/3 but standard Logistic distribution has varianceΠ²/3.
3. Moment generating function of standard Logistic distribution is $$M_z (t) \ = \ \Gamma(1-t) \ \Gamma(1+t) \ \ \ ; \ -1 \ < \ t \ < 1$$
4. The Logistic curve is symmetrical about the line Z = 0.

Uses of Logistic Distribution:

The Logistic distribution is mostly used as growth function in population and demographic studies and in time series analysis.

Properties of Negative Exponential Distribution:

• Mean of negative exponential distribution Expo (θ) is 1/θ.
• Variance of negative exponential distribution is1/θ².
• The r^th moment about origin of Expo (θ) is given by $$\mu_r^{'} \ = \ \frac{r!}{\theta^r} \ \ \ ; \ r \ = \ 1, \ 2, \ 3, \ 4$$
• In negative exponential distribution Expo (θ), $$if \ \theta \= \ 1, \ mean \ = \ variance$$ $$if \theta \ > \ 1, \ Mean \ > \ variance$$ $$if \ \theta \ < \ 1, \ mean \ < \ variance$$
• Moment coefficient of skewnessβ₁= 4⇒γ₁ = 2.
• Moment coefficient of kurtosisβ₂ = 9⇒γ₂ = 6. Thusthe negative exponential distribution is positively skewed and leptokurtic.
• The median of Expo (θ) is ln2/θ.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.