Mathematics Stack Exchange
Mathematics Stack Exchange

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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Missing $\lambda$ in Cauchy distribution

I have to following equality in my lecture notes $$\frac{1}{\pi \lambda} \int_{-\infty}^x \mathrm{arctan}'\left(\frac{y-\mu}{\lambda}\right)dy = \frac{1}{\pi} \int_{-\infty}^{\frac{x-\mu}{\lambda}} \mathrm{arctan}' \left(y\right)dy $$ and I…
mwater
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can I distort a von mises-fisher distribution?

I've been doing a bunch of work as of late with data that is well-described by (mixtures of) von Mises-Fisher (vmf) distributions: $$ \mathscr{F} \left(x \, \lvert \, \mu, \lambda \right) = C_D \left(\lambda \right) e^{\lambda \left<\mu,…
Matt
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The probability of missing a phone call

If Phone calls are received to a switch board at a rate of four calls per hour. If the operator leaves his workstation for half an hour what is the probability he will miss a phone call? I'm also confused to figure whether its related to Poisson…
ADG
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Proving probability density function of sigma notation

I have a question that is WAY over my head, and I wanted to ask any kind of help. Suppose $f_i(x)= 1, · · · , n$ are PDFs. That is, $f_i(x)≥0$ and $\int f_i(x)dx=1$ for all $i=1, · · · , n$. Consider a function $$g(x)=\sum_{i=1}^n p_i f_i(x)$$ where…
Kevin Kwon
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Find the distribution of $ X $ and $X + Y$

$f(x,y) = \exp(-2(x+y))$ , $0
Stephen
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What is distribution of $Z^2$ if$Z$ is the sum of a Gaussian and Rayleigh variable?

Let $Z=X+Y$; where $X\sim \mathscr N(0,\sigma^2_1)$ i.e. a Gaussian random variable and $Y$ follows the Rayleigh distribution: $$ f_Y(y) = \frac{y}{\sigma^2_2}\exp\left(-\frac{y^2}{2\sigma^2_2}\right) \mathbf{1}_{y \geqslant 0} $$ If we convolve…
debasish Bera
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How to calculate size distribution of an object than has been divided into smaller pieces

I have an object that is broken into smaller pieces such that the sum of the pieces is the volume of the original object. I have arranged the volumes from the smallest to largest and have calculated the cumulative volume starting at 0% and ending at…
Andrew
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How to compute the CDF of $X\cdot Y$ if $X,Y$ are independent and uniformly distributed over $(-1,1)$?

Just like in the title: How to compute the CDF of $X\cdot Y$ if $X,Y$ are independent random variables, uniformly distributed over $(-1,1)$? I tried using the next formula: the density of $X\cdot Y$ is the integral of $f(u/v)\cdot g(v)$ where…
ale
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What is the distribution of sum of a Gaussian and a Rayleigh distributed independent r.v.?

Let $Z=X+Y$; where $X\sim \mathscr N(0,\sigma^2_1)$ i.e. a Gaussian random variable and $Y$ follows the Rayleigh distribution: $$ f_Y(y) = \frac{y}{\sigma^2_2}\exp\left(-\frac{y^2}{2\sigma^2_2}\right) \mathbf{1}_{y \geqslant 0} $$ What will be…
debasish Bera
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What is the delta t on the conditional probability of failure (TTF)

Can someone explain me what is the $\Delta$t of the conditional probability of failure (CPF) on hazard function? CPF = P { t $\le$ T $\le$ t+$\Delta t$} $\mid$ T > t} $$CPF = \frac{R(t) - R(t- \Delta t )}{R(t)}$$
Andre Oliveira
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the probability that a chi square distribution smaller than its degree of freedom

Suppose $X$ is a $\chi_k^2$-distributed random variable, then is there any explicit form for the probability $$\mathbb{P} (X < k)?$$ In particular, I'm interested in the asymptotic value of $$\lim_{k\to \infty}\mathbb{P} (X < k). $$ Thank you.
Sean
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Importance Sampling

Suppose $p(x)$ approximates the density of interest $q(x)$. Then $$\int f(x) q(x) = \int f(x) \left(\frac{q(x)}{p(x)} \right) p(x) \ dx = E_{p(x)} f(x) \left(\frac{q(x)}{p(x)} \right)$$ Why don't the $p(x)$'s cancel in the second equality?
Mark Damien
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Probabilities of errors in three independent transmissions

i have been working through some old exam papers and have gotten stuck on this last one. can anyone help? When a piece of information (a bit) is transmitted over a communications channel, it may be wrongly communicated. One method of improving…
Rachael
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Functions of a random variable

Assume that $Y$ ~ $Exp(Ω)$. Find the cdf and pdf of $Z$ = |$Y$ - $δ$|. In order to solve this question so far, for $Y$, I am thinking about using the pdf equation for the exponential distribution i.e. $f(Y)$ = $Ωexp(-Ωx)$ for $x$ > $0$ and $0$ for…
Tera Peo
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Name for distributions for which all members of the family can be expressed as a transform of a member

Suppose one has a conventionally parameterized Normal Distribution in which the first parameter is a and the second parameter is b. Such a distribution can be expressed as a transform of the base Normal Distribution in which the first parameter is…
mathlawguy
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