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Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

Any probability distribution, including beta, binomial, chi, Erlang, gamma, geometric, lognormal, negative binomial, normal (Gaussian), Pareto, Poisson, Student's t, uniform, Wald, Weibull, zeta, and Zipf.

28080 questions
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Why use infimum in definition of Quantile function

I looked up the definition of Quantile function in Wikipedia, it is said that: The Quantile function is $Q(p)=\inf\{x\in R:p\le F(x)\}$ for $F:R\to(0,1)$, for a probability $0
user71346
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Distribution of the maximum of noisy random variables

Let $X_1, \ldots, X_N$ be $N$ hidden random iid variables, all with the same standard distribution, let's say uniform $\mathcal{U}(0, 1)$ or Gaussians $\mathcal{N}(0, 1)$ (probably easiest). I observe $N$ corresponding 'noisy' variables $Z_n = X_n +…
cdubout
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Let $X \sim \text{Unif}(0,1)$, what is the density function of $X (1-X)$?

I tried answers from https://stats.stackexchange.com/questions/21549/how-to-add-two-dependent-random-variables, however cannot solve my problem. I guess it is $$f_{X, -X^2}(x, y) = \begin{cases} 1, & 0 \leq x \leq 1, y = -x^2; \\ 0, &…
InterestedStudent
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Is there a known distribution for multinomial without replacement?

I would like to know if there's a known distribution for a multinomial sampling with limited bin size, or equivalently without replacement. The situation would be that I have $N$ bags of candy holding $k_i, i\in 1..N$ candies within them. Assuming I…
shians
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Conditional distribution of subvector of a Dirichlet random variable

Suppose $\mathbf{X} = (X_1, \cdots, X_K)$ follows a Dirichlet distribution with parameters $(\alpha_1, \cdots, \alpha_K)$. Partition $\mathbf{X} = (\mathbf{X}_{(1)}, \mathbf{X}_{(2)})$ for subvectors $\mathbf{X}_{(1)}, \mathbf{X}_{(2)}$. What is the…
Tom Chen
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Four-parameter Beta distribution and Wikipedia

Sorry if it is not an appropriate place for such questions, but anyway can anybody please confirm that the formula for the density function of the four-parameter Beta distribution is correct in Wikipedia. It seems $(c - a)$ is missing in the…
Ivan
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How to generate a random number from a pareto distribution

I'm working on a problem where I am trying to generate a random number from a Pareto distribution. Using some measured data, I have been able to fit a Pareto distribution to this data set with shape/scale values of $4/6820$ using the R library…
Jonathan Dunne
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what is the difference between joint probability distribution and random vector

Let $(S,\mathcal A, P)$ be a probability space and $\mathbf X:S\rightarrow \mathbb R^n$ random vector. Let $X_i:S\rightarrow \mathbb R$ be random variables such that $\mathbf X=(X_1,\ldots ,X_n)$. Is there any difference between distribution of…
Burt Named
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Median of binomial distribution

Intuitively, it makes sense that if $X\sim B(n,p)$, assuming that $np\in \mathbb{Z}$, then $P(X\leq E[X])\geq 1/2$. Wikipedia confirms this in the median section https://en.wikipedia.org/wiki/Binomial_distribution#Median. However, the results there…
Jia Cheng Sun
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Finding the distribution of a random variable with Laplace-Stieltjes transforms

In an exercise series from my Queueing Theory course I am asked to find $E(W)$, $P(W > 0)$ and $P(W > 1)$ where $W$ is the waiting time in a $M/G/1$ queue. In this exercise, the interarrival times $A$ are i.i.d. and exponentially distributed with…
Stijn
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How many finite sized, randomly placed holes on a finite area can be added before the number of holes decreases?

When adding finite sized, circular, randomly placed holes on a finite area the number of holes increases with each addition. After a number of circular holes have been added, adding another makes the new hole and an existing hole merge and the total…
D Duck
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I'm trying to identify a distribution presented to me as the "Van Loon distribution".

My professor's book on Electrical Measurements presents 3 distributions that model the probability density function of the error an observation is expected to have given a set of already obtained observations. These are the normal distribution, the…
gjl
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Expected maximum number of stones in a box

We have $N$ boxes and $N$ stones. We take a stone, put it into a randomly selected box, proceed to the next stone. In the end all the stones are in the boxes. Some boxes may be empty, some may contain several stones. Let $n$ be a maximum number of…
lesnik
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"General" non centered Chi distribution (having correlated random variables)?

Let $\mathbf{X} = [X_0, X_1]^t \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = [\mu_0, \mu_1]^t \in \mathbb{R}^2$ and $ \boldsymbol{\Sigma} = \begin{bmatrix} \sigma_0^2 & \rho\sigma_0\sigma_1 \\[0.3em] …
user49546
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How to write a proper definition of the uniform distribution on unit sphere?

If $X=(x_1, \ldots, x_n) \in \mathbb{R}^n$, then $x$ is on the unit sphere, if $\lVert X \rVert_{2}=\sqrt{x_1^2+\ldots+x_n^2}=1$. Now how can I write a proper definition of the uniform distribution on the unit sphere? I was checking the internet for…
user299124
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