# Questions tagged [gamma-distribution]

For problems that are related to gamma-family probability distributions.

283 questions
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### Using the Central Limit Theorem to form CI from Gamma distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. I am doing a question in my note, not sure am I doing correct or not....
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### Exact distribution of MLE exponential distribution

Let $y_1, \dots,y_n$ be i.i.d. random variables from $Exp(\theta)$, where $\theta$ is scale parameter. I've found the MLE $$\hat \theta=\frac{\sum^{n}_{i=1}y_i}{n}$$ Now I need to find the exact ...
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### Finding P(X<Y) of 2 random variables that are gamma distributed

X ~ Gamma ($r_1$ = 3, $\lambda_1$ = 2) Y ~ Gamma ($r_2$ = 5, $\lambda_2$ = 7) Find P(X > Y) To find this, I know that I can double integrate the joint distribution of X and Y but I wanted to know a ...
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### Adding 1 minute to Gamma Distribution

I am doing a simulation, and as an input, I need to add 1 more minute to the already existing Gamma Distribution that I have (Alpha = 2.58 and Beta = 24.5). What would my new alpha and beta values be? ...
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### customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
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### Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant)...
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### Finding the survival and distribution function of a system.

We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter. We also have $P(U_2 \leq x)=x^2$ ...
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### Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.

Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large. What I have come up with so far is: Let $X=$ the sum of all $X_i$, ...
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### Markov chain - distribution of probability of state at generic step

Let $S$ be a finite discrete state set. Let $X(i) \in S, i = 1,2, \ldots$ be a random variable sequence. I've built-up a Markov transition matrix from a set of sequences of states. State $s_1 \in S$ ...
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### How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf?

I have some problems trying to prove the following problem: A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf ...
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### Distribution of a weighted gamma random variable

I have a gamma distributed random variable $x$ with pdf $$p_x(x)=\frac{\lambda^r}{\Gamma(r)}x^{r-1}\exp(-\lambda x),$$ where $r$ and $\lambda$ are shape and rate parameters respectively. If for ...
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### Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find $f_X(x)$.

Question: Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find join density $f_{X,Y}(x,y)$ and find the probability density function of $X$ (simplify until there are no ...
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### Concerning Waiting Times for a Poisson Process and the Gamma Distribution

The Statement of the Problem A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate $2.5$ per year, and that an individual dies when $196$ ...
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### chi squared distribution of independent normal distributions that are not standard normal

I've been working on the following problem. I'm a bit confused about some of the specifics of how to arrive at the correct answer. I hope someone here could point me in the right direction: A dart ...
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### Calculate probability that container will be empty over 10min period?

Meteorologists model the volume of rain water collected over a given time interval, as a realisation of a compound stochastic process in which the arrival of raindrops is modelled as a realisation of ...
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### Entropy of gamma-exponential compound distribution

Following this question, I have the PDF of a gamma-exponential compound distribution as $$f(y) = \frac{\alpha\beta^{\alpha}} {(y+\beta)^{(\alpha+1)}}$$ For my application I need the entropy of ...
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### Question on gamma distribution and waiting times.

Let $X_1, X_2,\dots$ be iid. random variables each with density $xe^{-x}$ for $x > 0$ and $0$ otherwise. Let $S_0 = 0$ and $S_n = X_1 +\cdots + X_n$, and $N(t) = \max\ \{n : S_n < t\}$. I need ...
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### Unit Measure Axiom for the Gamma Distribution

I'm studying basic probability and in my lecture notes, it shows how the Gamma function results from the convolution of two exponential random variables. To introduce the gamma function it shows that ...
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### Transforming Gamma into Chi-squared distribution

My question is about how to transform a Gamma distrbution in a Chi-squared one. I know that $X\sim\Gamma(\frac{\nu}{2},2$) is the same as $X\sim \chi^2 (\nu)$. Thus, for example in a case in which I ...
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### Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
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### Which parameter should be considered as “scale” parameter for Gamma distribution?

I originally posted this question on crossvalidated. In case it would be considered too "nerdy" or useless there, I also posted it here with the hope to get more replies. From Wikipedia and probably ...
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### Sum of $N$ Gamma distributed random variables being $N$ a Gamma distribution random variable

Thanks in advance. Let $X$ a gamma-distributed random variable having scale $θ$ and shape $k$: $$X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta)$$ with its probability density function ...
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### Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can ...
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### Distribution of the difference between two random variables

I have two independent random variables of Erlang distribution or you can consider them Gamma distributions but they are positive. Z = Y-X The difference between them should be a distribution that ...
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### Sum of two independent exponentially distributed random variables

Suppose that the time between calls from your aunt Debie has an exponential distribution with a mean time of 3 days. What is the probability that you will get two calls in less than 3 days? So trying ...
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### Expected value of minimum of $n$ Gamma functions? [closed]

Suppose I have a set of $n$ Gamma functions, all with different parameters. I then draw one sample from each function. How can I find the expected value of the minimum of these samples?
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### Joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$ for independent exponential random variables $X_1$ and $X_2$
If $X_i$, $i=1,2$, are independent $gamma(\alpha_i,1)$ random variables. Find the joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$. I am trying to use transformation method to solve it. Let ...
### If $1\leq \alpha$ show show that the gamma density has a maximum at $\frac{\alpha - 1}{\lambda}$
So using this form of the gamma density function: $$g(t) = \frac{\lambda^{\alpha}t^{\alpha -1}e^{-\lambda t}}{\int_0^\infty t^{\alpha - 1} e^{-t} \, dt}$$ I would like to maximize this. Now i was ...