Gamma distribution
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Probability density function
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Cumulative distribution function
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| Parameters | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe shape (real)WikiTeX: wikitex.sh reported a failure, namely: I can't scribe scale (real)
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| Support | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| cdf | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Mean | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Median | |
| Mode | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe for WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Variance | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Skewness | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Kurtosis | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Entropy | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe WikiTeX: wikitex.sh reported a failure, namely: I can't scribe
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| mgf | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe for WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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| Char. func. | WikiTeX: wikitex.sh reported a failure, namely:I can't scribe
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In probability theory and statistics, the gamma distribution is a continuous probability distribution. For integer values of the parameter k it is also known as the Erlang distribution.
Contents |
[edit] Probability density function
The probability density function of the gamma distribution can be expressed in terms of the gamma function:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. is the shape parameter and WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. is the scale parameter of the gamma distribution.
(NOTE: this parameterization is what is used in the infobox and the plots.)
Alternatively, the gamma distribution can be parameterized in terms of a shape parameter WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. and an inverse scale parameter WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby., called a rate parameter:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
Both parameterizations are common because they are convenient to use in certain situations and fields.
[edit] Properties
The cumulative distribution function can be expressed in terms of the incomplete gamma function,
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
The information entropy is given by:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. is the polygamma function.
If WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. for WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. and WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. then
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. are independent.
The gamma distribution exhibits infinite divisibility.
If WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby., then WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby.. Or, more generally, for any WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. it holds that WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby.. That is the meaning of θ (or β) being the scale parameter.
[edit] Parameter estimation
The likelihood function is
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
from which we calculate the log-likelihood function
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. by taking the derivative and setting it equal to zero yields the maximum likelihood estimate of the WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. parameter:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
Substituting this into the log-likelihood function gives:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. by taking the derivative and setting it equal to zero yields:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. is the digamma function.
There is no closed-form solution for WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby.. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using Newton's method. An initial value of WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. can be found either using the method of moments, or using the approximation:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. then WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. is approximately
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
/usr/wikia/docroot/wiki.factory, baby.
which is within 1.5% of the correct value.
[edit] Generating Gamma random variables
Given the scaling property above, it is enough to generate Gamma variables with WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. as we can later convert to any value of β with simple division.
Using the fact that if WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby., then also WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby.,
and the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1],
then WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby..
Now, using the "α-addition" property of Gamma distribution, we expand this result:
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
,/usr/wikia/docroot/wiki.factory, baby.
I can't scribe /usr/wikia/docroot/wiki.factory, baby. are all uniformly distributed on (0, 1 ] and independent.
All that is left now is
to generate a variable distributed as WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. for WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby.
and apply the "α-addition" property once more. This is the most difficult part, however.
We provide an algorithm without proof. It is an instance of the acceptance-rejection method:
- Let m be 1.
- Generate WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
and WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
— independent uniformly distributed on (0, 1] variables./usr/wikia/docroot/wiki.factory, baby. - If WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
, where WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
, then go to step 4, else go to step 5./usr/wikia/docroot/wiki.factory, baby. - Let WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
. Go to step 6./usr/wikia/docroot/wiki.factory, baby. - Let WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - If WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
, then increment m and go to step 2./usr/wikia/docroot/wiki.factory, baby. - Assume WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
to be the realization of WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby.
Now, to summarize,
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
,/usr/wikia/docroot/wiki.factory, baby.
where
WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. is the integral part of α,
ξ has been generating using the algorithm above with WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. (the fractional part of α),
WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. and WikiTeX: wikitex.sh reported a failure, namely:I can't scribe /usr/wikia/docroot/wiki.factory, baby. are distributed as explained above and are all independent.
[edit] Related distributions
- WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is an exponential distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
for any c > 0 ./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is a gamma distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
and if the WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
are all independent and share the same parameter WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is a chi-square distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - If WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is an integer, the gamma distribution is an Erlang distribution (so named in honor of A. K. Erlang) and is the probability distribution of the waiting time until the WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
-th "arrival" in a one-dimensional Poisson process with intensity WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
then WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
, where WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
is the inverse-gamma distribution./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is a beta distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
< and WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
and are also independent./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is a Maxwell-Boltzmann distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
is a normal distribution as WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
where WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby. - The real vector WikiTeX: wikitex.sh reported a failure, namely:
I can't scribe
follows a Dirichlet distribution if WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
are independent, and WikiTeX: wikitex.sh reported a failure, namely:/usr/wikia/docroot/wiki.factory, baby.I can't scribe
./usr/wikia/docroot/wiki.factory, baby.
[edit] References
- R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)
[edit] See also
es:Distribución gammafi:Gamma-jakauma sv:Gammafördelning
| This page uses content from the English-language version of Wikipedia. The original article was at Gamma_distribution. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
