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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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Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if there is a closed expression for the density. Can…
Peter Lustiger
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Wasserstein distance between distribution functions

It is well-known fact that if we have two DFs F and G with finite second moments, then one can calculate the Wasserstein distance between them using this formula: $$ W_2^2(F,G) = \inf E(ξ-η)^2 = \int_0^1|F^{-1}(t)-G^{-1}(t)|^2 \, dt , $$ where…
Ilya68
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str = md5( str ), String equals hash value of the string

Is there a way to prove that a 32-byte string exist (or not) for which the MD5 hash function result is equal to the string itself ? str = md5( str ) Or can one say something about the probability of such a collision.
tur1ng
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Any good text on algebra of probability distribution functions?

I'm wondering whether there exists any good text on transformations of probability distribution functions. I do know how to get the probability-function of a function of a random-variable. I am interested in some handy book, which provides direct…
kaka
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Is there a name for the probability distribution with the form $p(x) = a \,x^2\, \exp( -b\,x^2 )$

There is a probability distribution: $$p(x) = a \,x^2\, \exp( -b\,x^2 ), \quad a,b>0,\ x \in ( -\infty,\,\infty ) $$ I wonder which probability distribution is it?
Lyn
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Poisson distribution given Gamma Distribution

I'm struggling with this one: If $\theta $ is a Gamma$(p,\lambda)$ random variable with $p>1$ and $\lambda>0$. We give the density of the gamma distribution: $ f(x) = \frac {\lambda^p}{\phi(p)} x^{p-1} \exp(-\lambda x) 1_{t>0}$ $\phi (p)$ is…
XCoder
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Generalized Laplace distribution?

definition: Laplace distribution $Lap(\mu, b)$ with mean $\mu$ and a scaling paramter $b$ is defined as $$f_X(x;\mu, b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right)$$ The standard Laplace distribution is a simplifed version where $\mu = 0$…
Federico Magallanez
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expected value and variance of the difference of number of people in a row.

I need to calculate the expected value and the variance of the following variable: $n$ people sit in a row, among them person 'a' and person 'b'. Define $X$ to be the amount of people between 'a' and 'b'. Calculate $E(X)$ and $Var(X)$. I have…
Jeff
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Is a log-normal distribution uniquely determined by its moments or not?

Wikipedia states that A log-normal distribution is not uniquely determined by its moments $\text{E}[X^k]$ for $k\ge 1$, that is, there exists some other distribution with the same moments for all $k$. In fact, there is a whole family of…
Řídící
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Transformation of probability density function under indicator function

I have to calculate the pdf $f_Y(y)$ where $y=\mathbb{I}_{\left[-c,c\right]}( x )$ where the pdf of $x$ is known and denoted by $f_X(x)$ and $c$ is a constant. In this case, $\mathbb{I}_{\mathcal{A}} ( x )$ denotes the indicator function and is $x$…
bonanza
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What Implications Can be Drawn from a Binomial Distribution?

Hello everyone I understand how to calculate a binomial distribution or how to identify when it has occurred in a data set. My question is what does it imply when this type of distribution occurs? Lets say for example you are a student in a physics…
user17321
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Confused about equating random variables, vs. their distributions

Let's say I have two random variables, $X$ and $Y$. I know the relationship between their CDF's: $$F_Y(y) = g(F_X(y))$$ For example, I could know that $F_Y(y) = F_X(y)^2$ or something like that. What then can I say about relating $Y$ to $X$? I am…
Angada
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Is there a name for this discrete probability distribution?

What's the name for this probability distribution over the positive integers? (if it has a name) P(n) = $\frac{1}{n} - \frac{1}{n+1}$ = $\frac{1} {n * (n+1)}$ Given a uniformly distributed real number r from 0 to 1, this is the distribution for…
dspyz
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Scaling a probability distribution function

I have the following PDF that gives the probability of a certain annual wage being drawn: $f(w)=0$ if $w<20000$ $\frac{w-20000}{50000^2}$ if $w \in [20000,70000]$ $\frac{120000-w}{50000^2}$ if $w \in (70000,120000]$ $0$ if $w>120000$ I want to scale…
Jess
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What's the $n$'th cumulant of the Beta distribution?

Is there a closed form for the $n$'th cumulant of the Beta distribution, as a function of the parameters $\alpha$, $\beta$? The cumulant generating function of the Beta distribution is the logarithm of a confluent hypergeometric…
a06e
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