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Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

In applied probability, a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:

  • by supplying a valid probability mass function or probability density function
  • by supplying a valid cumulative distribution function or survival function
  • by supplying a valid hazard function
  • by supplying a valid characteristic function
  • by supplying a rule for constructing a new random variable from other random variables whose joint probability distribution is known.

A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

See also: Wikipedia

1926 questions
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Uniform distribution on a simplex

In a context where I try to estimate some combinatorial sums, I'm faced with vector random variables $(x_1,...,x_n)$ uniformly distributed with $n \rightarrow \infty$. I want to know if the components have to behave in a wellknown fashion. I…
Gianfranco
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Distribution of dropped objects

Consider small perfectly elastic spheres being dropped from a fixed height in R^3, bouncing and coming to rest on the horizontal R^2. Assuming a reasonable distribution of minor perturbations of the initial velocity and minor perturbations of…
Jim Stasheff
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What does it mean to sample a value x* from f(x)?

This might be a really elementary question, but I'm not sure what it means. I have a density function f(x). How do I sample a value from f? For known distributions there are functions in R which do it for you (e.g. runif, rnorm, etc.) but how do I…
ysl
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Is this function monotonically increasing?

Suppose that $\boldsymbol{t}\sim \mathcal{N}(\boldsymbol{u};\boldsymbol{0},\boldsymbol{M})=f_{\boldsymbol{t}}(\boldsymbol{u})$, where $\boldsymbol{t}$ is a $N$-dimensional gaussian random vector,…
HiNull
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Central Limit Theorem when the sum of the variances is finite

Suppose $(x_i)_{i\in\mathbb{N}}$ a set of strictly positive numbers such that $L=\sum_{i\in\mathbb{N}}x_i$ is finite. Suppose that $(X_i)_{i\in\mathbb{N}}$ is a set independant (real-valued) random variables, each uniformly distributed in…
ThibThib
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Distribution and moments of ratio of two beta variables?

If $X$ and $Y$ are two Beta random variables, I am interested in the distribution of their ratio $X/Y$. More specifically, I am interested in the moment generating function of this ratio. There is a paper of Pham-Gia that apparently computes the…
Aryeh Kontorovich
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Distribution of Maximum of a uniform multinomial distribution

Hello, I'm working with a data structure which uses a uniform distribution to bucket the inputs into $k$ buckets. The efficiency of the structure is bounded by the $\frac{k_{max}}n$, where $n$ is the number of items. How many elements are in each…
Jack
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on the difference of exponential random variables

Assume two random variables X,Y are exponentially distributed with rates p and q respectively, and we know that the r.v. X-Y is distributed like X'-Y' where X',Y'are exponential random variables, independent among themselves and independent of X…
Mensarguens
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CLT convergance rate for sum of log-normals

Hello, all! I have a big sum of log-normal (with location parameter $\mu$ and scale parameter $\sigma$) random variables $X_i$ $\sum_{i=1}^N X_i$ with $N \gg 1$. How could I estimate convergence rate to a gaussian distribution relative to $\mu$ and…
spk
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density of distance between points in unit circles

Let $a$ and $b$ be two points in the plane. Let's choose a point $c$ uniformly from the circle of radius $r$ with $a$ as center and choose a point $d$ uniformly from the circle of radius $r$ with $b$ as the centre of this circle. Let $X$ be the…
Mudi
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What's the probability of differences among n independent uniform distribution variables?

Given n independent random variables $x_1,x_2,...,x_n$, they have standard uniform distributions over [0,1]. Then what's the probability that there is at least one $|x_i-x_j| >= d$ for any different $i,j$ and $0<=d<=1$? The discrete form of this…
user42931
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Interpreting a non-integer shape parameter for the gamma distribution

An often-cited example of the gamma distribution is that for integer shape parameter k, and scale parameter lambda, the gamma distribution can be conceptualized as the sum of k independent (identically) exponentially-distributed random variables…
Yannick
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Joint distribution on order statistics and sample history

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the discrete uniform distribution $[1,2, ..., P]$, what is the joint distribution of the last $k$ order statistics and last $k$ samples:…
LNguy
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Formula for maximum of two Gumbel distributions?

I have parameters of two Gumbel distributions ($\mu_1, \beta_1)$ and $(\mu_2, \beta_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu_m, \beta_m$, so that: $Gumbel(\mu_m,\beta_m)$ = $max(Gumbel(\mu_1, \beta_1), Gumbel(\mu_2,…
reg
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