All Questions
Tagged with gamma-distribution density-function
45 questions
2
votes
0
answers
38
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Distribution supported on $(0,\infty)$ for which moments of its truncated distribution are elementary functions of the truncation point and power
I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\...
- 525
1
vote
1
answer
72
views
Conditional density following Gamma distribution
We know that if a random variable say x ~ Gamma(a, b), then its probability density function is $ \propto x^{a-1} exp^{-bx}$.
In a Bayesian hierarchical model, for example
$Z_1, \cdots, Z_n |\theta \...
- 13
4
votes
0
answers
89
views
What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?
Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of
$$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
- 141
1
vote
1
answer
103
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Issue about both results in agreement with 2 different ways to compute variance of a random variable : weighted chisquared vs Gamma distributions
1.) I am interested in computing the variance of this observable $O$ involving the coefficients of spherical harmonics $a_{\ell m}$ and the $C_{\ell}$ which is the variance of an $a_{\ell m}$ :
$$O=\...
user226073
1
vote
1
answer
321
views
Get probability distribution function from density function and calculate the cumulative value [duplicate]
For the given density function, how to find its distribution function and how to calculate the value of the distribution function?
Density function:
$$f(x) = \frac{1}{\Gamma(\frac{n}{2})}x^{\frac{n}{2}...
- 19
0
votes
0
answers
69
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PDF of the exp-gamma distribution
exp-gamma distribution is defined as the density of the random variable log(X) when X is a gamma random variable.
I am trying to obtain its PDF. Unfortunaltely, the only formula I have found is from a ...
- 566
0
votes
1
answer
92
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Random numbers with exponentiated gamma distribution? [closed]
How to get random numbers following "exponentiated gamma distribution"?
I tried to search some functions in R
and this is what i got:
https://rdrr.io/cran/Newdistns/man/expg.html
I want to ...
- 171
1
vote
1
answer
56
views
Properties of $\chi^2(1)$ multiplied by a real value "$a$"
Is it true that the $\chi^2$ distribution with $k=1$ (noted $\chi^2(1)$) multiplied by a real value $a$ is equal to $\chi^2(a)$ ?
If not, is there a particular distribution for $a\cdot\chi^2(1)$?
If ...
user226073
-1
votes
1
answer
205
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Consistency when we want to find the distribution of sum of random variables following each one a distribution
I want to clarify a point that disturbs me among different cases.
I am interested in formulate correctly in a general case when we know the distribution of different random variables and we want to ...
user226073
0
votes
1
answer
187
views
Scaled gamma random variable with threshold
Let's say we have gamma random variables X, which has $k$-shape parameter and $\theta$-scale parameter and threshold parameter $\alpha$ ($x>\alpha$).
What is the distribution of cX, is it still ...
3
votes
1
answer
440
views
Finding a right way of sampling 1/X knowing X follows the Moschopoulos distribution (sum of Gamma distribution with different (shape/rate parameters)
I can generate, with COGA R library (with rcoga function), a sample from a random variable ...
user226073
1
vote
1
answer
241
views
Conditional distribution using Gamma and Weibull
I'm trying to compute the conditional distribution of $X|Y = y$.
$X\sim Gamma(3,2)$
$Y|X = x \sim Weibull(2,x)$
I was doing this:
$f_{X|Y = y}(x) \propto f_x(x)\cdot f_{Y|X = x}(y)$
$f_x(x) = \frac{1}{...
- 69
1
vote
0
answers
33
views
PDF of function of gamma distribution
If $X = {X_1, \dots, X_n}$ is a sample of independent and identically distributed random variables, each following a Gamma distribution $\Gamma(2, λ)$, with unknown scale parameter $\lambda$, then, ...
- 11
1
vote
0
answers
65
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Proving that $\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=\sum_{x=0}^{r-1}\frac{e^{-\mu}\mu^x}{x!}$ [duplicate]
$$\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=\sum_{x=0}^{r-1}\frac{e^{-\mu}\mu^x}{x!}$$
What I have tried-
$$\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=e^{-\mu}\sum_{x=0}^{r-1}\...
- 184
0
votes
0
answers
38
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Unable to plot obtain realizations of Gamma distribution variant
I have the probability density function (pdf) for $\sigma^2 = 1/\phi$, where $\phi\sim \text{Gamma}(a, b)$. I am trying to simulate 1000 realizations of $\phi$ and then plot a histogram using the pdf ...
- 654
0
votes
0
answers
133
views
Ratio distribution of variance-gamma distribution and normal distribution
How does one calculate the probability density function of a random variable $Z$, defined as the ratio distribution of a variance-gamma distribution $X$ and a normal distribution $Y$ i.e. $Z=X/Y$?
The ...
- 1
0
votes
1
answer
448
views
Transformation of a random variable with a gamma distribution
Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$.
But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
- 239
1
vote
1
answer
119
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Divide beta from a gamma distribution to get another gamma distribution?
In the textbook, there's a distribution like the following,
$S=\sum_{i=}^{200}X_i\sim Gamma(\alpha = 200, \beta)$
then the textbook define a new function $P$ obtained by diving the $\beta$, so ...
- 309
1
vote
0
answers
107
views
Finding the distribution of a piecewise function of a Gamma random variable
Let random variable $X \sim \text{Gamma}(\alpha,\beta)$. I want to derive the distribution of $Y$, where:
$$
Y = \left\{
\begin{array}{ll}
a X - k & \quad X \geq \frac{k}{a} \...
- 11
2
votes
0
answers
173
views
Characterizing a distribution
I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year.
If I have some data in the form ...
- 123
2
votes
1
answer
651
views
When a probability density function is defined to be finite?
In "Pattern recognition and machine learning" by Cristopher Bishop, Chapter 2.3.6 (pag. 100) says that
The gamma distribution has a finite integral if $a>0$, and the
distribution itself is ...
- 2,433
0
votes
1
answer
62
views
Probability question in Mat
My teacher give me this question:
Using MATLAB, generate 10000 Random Vectors of size 500 with the PDF of Gamma distribution. Find the PDF of maximum and minimum of the generated Random vectors.
(Use ...
- 1
1
vote
1
answer
969
views
finding quantiles of a kernel density estimation
I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step).
I used code density()...
- 11
0
votes
2
answers
3k
views
Confused about Gamma distribution parameters in R
I would like to draw a Gama distribution in R but Im confused since it has different notations.
Mine is in the form:
$ \theta^\alpha x^{\alpha-1}e^{-\theta x} / \Gamma(\alpha) $
Let's say for ...
- 247
0
votes
1
answer
1k
views
Gamma(1,1/θ) interchangeable with exp(1/θ) and exp(θ)?
Am I correct that the probability density function:
$f(x;\theta)=\theta e^{-\theta x}$
is $Gamma(1, 1/\theta)$ or $exp(1/θ)$ or $exp(θ)$ ?
They all have the same distribution?
- 73
3
votes
2
answers
665
views
Merits of reparameterizing the Gamma and inverse Gamma
Wikipedia states that the PDFs for the Gamma distribution is:
$$
f(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x)
$$
However, in Rasmussen 2000, the pdf for the ...
- 345
6
votes
2
answers
6k
views
Deriving exponential distribution from sum of two squared normal random variables
Let $X$, $Y$ be i.i.d. random variables with distribuition $\mathcal{N}(0,1/2)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.
1
vote
0
answers
214
views
Transformation of two gamma variables
If you have $\frac{X}{X+Y}$ where X and Y are both independent Gamma distributions where $\alpha$ for both X and Y is different, but $\beta$ is the same for both, then how would that be different from ...
- 15
6
votes
1
answer
419
views
Sum of truncated Gammas
I have a set of i.i.d. variables $X_i$ that are distributed according to a truncated $\text{Gamma}(\alpha,\beta)$ distribution, with support on $[0,w)$ where $w$ is a known constant. What's the ...
- 535
4
votes
0
answers
207
views
Sum of truncated Gammas and degenerate
I have a variable $X$ which I am modelling with a mixture model:
$$\begin{aligned}
(X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\
(X|B) &...
- 535
3
votes
0
answers
122
views
Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?
I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by
$$ \pi_G(\xi|\mu,\sigma) = {1\...
- 724
0
votes
1
answer
2k
views
PDF and CDF of sum of two independent $\Gamma$-distributed random variables [duplicate]
Let $X \sim \Gamma(m, p)$ with a shape parameter $m$ and a scale parameter $p$ and $Y \sim \Gamma(m, q)$ with a shape parameter $m$ and a scale parameter $q$, and let $X$ and $Y$ be independent.
...
4
votes
0
answers
8k
views
Relationship between the Gamma and Beta distributions [duplicate]
I was looking at a proof of the following fact
Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then
$$
...
- 1,465
4
votes
1
answer
395
views
Moment generating function of a distribution
I want to find the moment generating function (mfg) and mean deviation of this distribution:
$$f(x,\epsilon,k,\theta) = k\theta^{(1+1/k+\epsilon/k)}x^{(k+\epsilon)}\exp{(-\theta x^k )}/(\Gamma(1+(1+\...
- 71
3
votes
2
answers
645
views
Identifying distribution of a variable
Consider a variable that can take both negative and positive values, and that has the following density plot:
I am trying to identify the distribution of this variable. The density plot resembles ...
- 957
5
votes
1
answer
8k
views
Why does fitdistr does not work with gamma?
I am tring to find a probability density for some waiting time, but I am having a hard time. Fitdistr does not work with Gamma. Am I missing something?
Is there ...
- 213
2
votes
2
answers
154
views
Ordered gamma variables led to an ugly integral
Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f.
$$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$
and let $Y_1,...,Y_n$ be the order statistics for these variables.
a) Find the conditional p. ...
- 425
52
votes
2
answers
36k
views
Gamma vs. lognormal distributions
I have an experimentally observed distribution that looks very similar to a gamma or lognormal distribution. I've read that the lognormal distribution is the maximum entropy probability distribution ...
- 1,257
42
votes
4
answers
17k
views
Good methods for density plots of non-negative variables in R?
plot(density(rexp(100))
Obviously all density to the left of zero represents bias.
I'm looking to summarize some data for non-statisticians, and I want to avoid ...
- 13.7k
2
votes
1
answer
2k
views
Fitting data to gamma distribution to find score which corresponds to pvalue < 0.05?
I have data of size 116.667 rows defined as:
...
- 21
19
votes
3
answers
46k
views
The sum of two independent gamma random variables
According to the Wikipedia article on the Gamma distribution:
If $X\sim\mathrm{Gamma}(a,\theta)$ and $Y\sim\mathrm{Gamma}(b,\theta)$, where $X$ and $Y$ are independent random variables, then $X+Y\sim ...
- 193
13
votes
1
answer
10k
views
Density of Y = log(X) for Gamma-distributed X
This question is closely related to this post
Suppose I have a random variable $X \sim \text{Gamma}(k, \theta)$, and I define $Y = \log(X)$. I would like to find the probability density function of $...
- 555
9
votes
1
answer
6k
views
Maximum Likelihood Estimation of Inverse Gamma Distribution in R or RPy
I am trying to fit a three parameter inverse gamma distribution to my data in either R or Python. I would like to do this using maximum likelihood estimation (MLE).
The pdf of the three parameter ...
- 137
35
votes
4
answers
205k
views
Sum of exponential random variables follows Gamma, confused by the parameters
I've learned sum of exponential random variables follows Gamma distribution.
But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what ...
- 353
6
votes
2
answers
4k
views
Bivariate Gamma distribution PDF
I'm analyzing a set of data, and I like to fit a gamma distribution. I know how to do it in one dimension, but the data that I'm analyzing now are two dimensional. Is there any way that I can have a ...
- 61