The general formula for the
of the gamma distribution is
\( f(x) = \frac{(\frac{x-\mu}{\beta})^{\gamma - 1}\exp{(-\frac{x-\mu}
{\beta}})} {\beta\Gamma(\gamma)}
\hspace{.2in}
x \ge \ \gamma,
\beta > 0 \)
where & is the ,
& is the ,
& is the , and &G
is the gamma function which has the formula
\( \Gamma(a) = \int_{0}^{\infty} {t^{a-1}e^{-t}dt} \)
The case where & = 0 and & = 1 is called the
standard gamma distribution.
The equation for the standard gamma
distribution reduces to
\( f(x) = \frac{x^{\gamma - 1}e^{-x}} {\Gamma(\gamma)}
\hspace{.2in}
x \ge 0; \gamma > 0 \)
Since the general form of probability functions can be
, all subsequent formulas in this section are
given for the standard form of the function.
The following is the plot of the gamma probability density function.
The formula for the
of the gamma distribution is
\( F(x) = \frac{\Gamma_{x}(\gamma)} {\Gamma(\gamma)}
\hspace{.2in}
x \ge 0; \gamma > 0 \)
where &G is the gamma function defined above and
\(\Gamma_{x}(a)\) is the incomplete gamma function.
The incomplete gamma
function has the formula
\( \Gamma_{x}(a) = \int_{0}^{x} {t^{a-1}e^{-t}dt} \)
The following is the plot of the gamma cumulative distribution
function with the same values of & as the pdf plots above.
The formula for the
of the gamma distribution does not exist in a simple
closed form.
It is computed numerically.
The following is the plot of the gamma percent point function with
the same values of & as the pdf plots above.
The formula for the
of the gamma distribution is
\( h(x) = \frac{x^{\gamma - 1}e^{-x}} {\Gamma(\gamma) -
\Gamma_{x}(\gamma)}
\hspace{.2in}
x \ge 0; \gamma > 0 \)
The following is the plot of the gamma hazard function with the same
values of & as the pdf plots above.
The formula for the
of the gamma distribution is
\( H(x) = -\log{(1 - \frac{\Gamma_{x}(\gamma)} {\Gamma(\gamma)})}
\hspace{.2in}
x \ge 0; \gamma > 0 \)
where &G is the gamma function defined above and
\(\Gamma_{x}(a)\) is the incomplete gamma function defined above.
The following is the plot of the gamma cumulative hazard function with
the same values of & as the pdf plots above.
The formula for the
of the gamma distribution is
\( S(x) = 1 - \frac{\Gamma_{x}(\gamma)} {\Gamma(\gamma)}
\hspace{.2in}
x \ge 0; \gamma > 0 \)
where &G is the gamma function defined above and
\(\Gamma_{x}(a)\) is the incomplete gamma function defined above.
The following is the plot of the gamma survival function with the same
values of & as the pdf plots above.
does not exist in simple closed form.
It is computed
numberically.
The following is the plot of the gamma inverse survival function with
the same values of & as the pdf plots above.
Common Statistics
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.
&&&& & & 1
0 to \(\infty\).
Standard Deviation
\( \sqrt{\gamma} \)
\( \frac{2} {\sqrt{\gamma}} \)
\( 3 + \frac{6} {\gamma} \)
Coefficient of Variation
\( \frac{1} {\sqrt{\gamma}} \)
Parameter Estimation
The method of moments estimators of the 2-parameter gamma distribution
\( \hat{\gamma} = (\frac{\bar{x}} {s})^{2} \)
\( \hat{\beta} = \frac{s^{2}} {\bar{x}} \)
\(\bar{x}\) and s are the sample mean and standard
deviation, respectively.
The maximum likelihood estimates for the 2-parameter gamma
distribution are the solutions of the following simultaneous
\( \hat{\beta} - \frac{\bar{x}}{\hat{\gamma}} = 0 \)
\( \log{\hat{\gamma}} - \psi(\hat{\gamma}) - \log \left( \frac{\bar{x}}
{ \left( \prod_{i=1}^{n}{x_i} \right) ^{1/n}
} \right) = 0 \)
with & denoting the digamma function.
These equations need to be
this is typically accomplished by using statistical
software packages.
Some general purpose statistical software programs support at least
some of the probability functions for the gamma distribution.<meta name="keywords" content="生日,渠道商,DQ,冰雪皇后,惠普电脑,网站,退款,消费者,DQ,过期,
团购退款服务不过是在作秀_IT频道_新华网
团购退款服务不过是在作秀
日 13:41:19
中国计算机报
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　　头顶“中国首家团购网站”光环的美团近日迎来上线一周年。但这个“生日”王兴过得并不轻松。2月末，美团推出了DQ冰雪皇后现金券，但与美团签订合作协议的是DQ公司的某一渠道商，而非DQ公司本身。因低价协议损害了其他渠道商的利益，DQ公司以合同不真实为由宣称此次活动无效。
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　　2010年央视3·15晚会曝光惠普电脑质量门之后，惠普电脑在中国的市场份额从17%下跌至不到10%。如今，3·15再次临近，据称此次3·15晚会的曝光重点将是以团购网站为主的电子商务企业。团购网站纷纷在3·15晚会之前推出各种退款服务，与其说是尊重消费者权益、促进行业规范化发展，倒不如说是为了应对3·15晚会而提前进行的危机公关。
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