# 数学代写|概率论代写Probability theory代考|STAT4061

## 数学代写|概率论代写Probability theory代考|LOGLOGISTIC DISTRIBUTION

A random variable $X$ is said to have loglogistic distribution if its pdf $f(x)$ is of the following form.
$\mathrm{f}(\mathrm{x})=\frac{\alpha x^{\alpha-1}}{\left(1+x^\alpha\right)^2}, \mathrm{x} \geq 0, \alpha$ is a positive real number.
We will denote the loglogistic distribution by $\mathrm{LL}(\alpha)$ if its $p d f$ is as given above.
The Figure 4.10. gives the pdf of $\mathrm{LL}(2), \mathrm{LL}(4)$ and $\mathrm{LL}$ (8).

Mean $=\frac{\pi}{\alpha} \csc \frac{\pi}{\alpha}, \alpha>1$.
Variance $\left.=\frac{2 \pi}{\alpha} \csc \frac{2 \pi}{\alpha}\right)-\left(\frac{\pi}{\alpha} \csc \frac{\pi}{\alpha}\right)^2 \cdot \alpha>2$.
$\mathrm{E}\left(X^k\right)=\cdot \frac{k \pi}{\alpha} \csc \frac{k \pi}{\alpha}, \alpha>k$
Moment gencrating function $\mathrm{M}(\mathrm{t})=\sum_{k=0}^{\infty} \frac{t^k}{k !} B\left(\frac{\alpha+k}{\alpha}, \frac{\alpha-k}{\alpha}\right)$
Characteristic function $\varphi(y)=\sum_{k=0}^{\infty} \frac{(i t)^k}{k !} \bar{B}\left(\frac{\alpha+k}{\alpha}, \frac{\alpha-k}{\alpha}\right)$.

## 数学代写|概率论代写Probability theory代考|LOGISTIC DISTRIBUTION

The $\operatorname{pdf} \mathrm{f}(\mathrm{x})$ of the logistic distribution is
$$\mathrm{f}(\mathrm{x})=\frac{1}{\sigma} \frac{e^{-\frac{x-\uparrow \mu}{\sigma}}}{\left(1+e^{\left.-\frac{x-\rho}{\sigma}\right)^2}\right.},-\infty<x-\mu<\infty, \sigma ? 0 .$$
We denote the logistic distribution with the above pdf as $\operatorname{LO}(\mu, \sigma)$. The graphs of the pdfs of LO (1/2), LO (1) and LO (3) are given in Figure 4.11.

Mean $=\mu$
Variance $=\frac{\pi^2 \sigma^2}{3}$
$\mathrm{E}(x-\mu)^{2 m+1}=0, \mathrm{~m}=1,2,3, \ldots$
$\mathrm{E}\left((X-\mu)^{2 m}=2 \sigma^{2 m}\left(2^{2 m-1}-1\right) B_m, \mathrm{~m}=1,2.3, \ldots\right.$.

where $\bar{B}_m$ is the mth Bernoulli number.
Moment generating function $\mathrm{M}(\mathrm{t})=e^{\mu t} \Gamma(1-\sigma t) \Gamma(1+\sigma t)$,
Characteristic function $\varphi(t)=e^{\mu i t} \Gamma(1-\sigma i t) \Gamma(1+\sigma i t)$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|LOGLOGISTIC DISTRIBUTION

$\mathrm{E}\left(X^k\right)=\cdot \frac{k \pi}{\alpha} \csc \frac{k \pi}{\alpha}, \alpha>k$

## 数学代写|概率论代写Probability theory代考|LOGISTIC DISTRIBUTION

$$\mathrm{f}(\mathrm{x})=\frac{1}{\sigma} \frac{e^{-\frac{x-\uparrow \mu}{\sigma}}}{\left(1+e^{\left.-\frac{x-\rho}{\sigma}\right)^2}\right.},-\infty<x-\mu<\infty, \sigma ? 0 .$$

$\mathrm{E}(x-\mu)^{2 m+1}=0, \mathrm{~m}=1,2,3, \ldots$
$\mathrm{E}\left((X-\mu)^{2 m}=2 \sigma^{2 m}\left(2^{2 m-1}-1\right) B_m, \mathrm{~m}=1,2.3, \ldots\right.$

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