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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Push-forward of gaussian meaesure onto unit ball

I am trying to find a function $\phi$ such that for $Z \sim \mathcal{N}(0, I_n)$, which denotes the multidimensional gaussian distribution, we have that $\phi(Z) \sim Uniform(B_n^1)$, where $B_n^1$ is the unit ball. For the unit cube, $\phi$…
JohnKnoxV
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Calculating density of function of random variable

$X\in Un(0,1)$, form $U=-ln(1-x)$. I want to calculate $f_U$. My attempt: $F_U(t)=P(U\leq t)=P(-ln(1-X)\leq t)=P(1-X \geq e^{-t})=P(X\leq 1-e^{-t})=F_X(1-e^{-t})$. Since $X\in Un(0,1)$. It has distribution $F_X(t)=\begin{cases} 0 &t<0 \\t & 0\leq t…
fejz1234
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What distribution has these properties?

There is a 2D continuous distribution from an actual physical process which I want to model mathematically. I have 40k samples. You can see a histogram here and here. This can be summarized as 4 properties, listed in order of prevalence. Further…
EPICI
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Find the probability density $f_Y(y)(Y= X_1 + X_2 + 1)$

Problem: Let $X=[X_1,X_2]^T$ be a real-valued Gaussian random vector with mean and co-variance $$\mu:=[1, 2]^T\,\,\,,\,\,\, \Lambda := \begin{pmatrix}2&-1\\-1&3\end{pmatrix}$$ Find the probability density $Y = AX + B$, where $A := [1, 1]$ and $B :=…
Mr.Robot
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What exactly is min/max in probability?

I don't quite understand what the min and max functions do in terms of probability and random variables or how they're able to have PDFs and CDFs. If $V = max(X_1, X_2 ... X_n) $ , what is getting returned? Is it supposed to be the max value of each…
AWW
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How to quantify skewness in a skewed Gaussian distribution?

I am looking at the wikipedia page for the skewed Gaussian distribution, and am interested in quantifying the skewness. From the first subsection, I gather that the theoretical maximum skewness $\gamma_1$ can only be $.9952717...$. I am working with…
paulinho
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Product distribution. Integration limits

I have some troubles with understanding of product distribution calculation. Consider a simple example: $f_X(x)$=$\frac{1}{2}$ for $1\le x \le 3$ and $f_Y(y)$=$\frac{1}{4}$ for $2 \le y \le 6$. Find $f_Z(z)$ if $Z=XY$. To find pdf of $Z=XY$, I'm…
user474290
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Determine the distribution that has all moments equal to $\mu$ where $ 0 < \mu < 1$

Determine the distribution that has all moments equal to $\mu$ where $ 0 < \mu < 1$ Having difficulty getting started on this, any help appreciated.
Rubicon
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Confusion between exponential and poisson parameters

I will write down a question that I'm confused about: The time elapsed between arrivals of customers is modeled that $V_k$ represents the time elapsed between the arrivals of the $(k-1)$th and $k$th customer. ($V_1 = $ the time the first customer…
similarityinvariance1
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Distribution with nothing but average

I am working on a discussion of the estate tax effect on farms. The data I have is there are about 2 million family (i.e. not large corporation) farms, and the average value is 1.2M dollars. On the passing of both owners, there would be a tax due if…
JTP - Apologise to Monica
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Triangular distribution question

I am trying to solve the following problem Theo the monkey and Colby the hippo have a 20-ounce cake. Colby cuts it in half and lets Theo pick the larger piece. But Colby is a hippo, so he can’t cut very well. In particular Colby cuts the cake into…
Dider
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Find the pdf, mean and distribution of a random variable given the mgf

Question: The moment generating function of the exponential distribution with rate $\alpha$ is given by $\dfrac{\alpha}{\alpha-s}$. Use this result to determine what distribution the random variable $Z$ has, where its mgf is $m_z(s) =…
Rubicon
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Finding the distribution of a random variable knowing the distribution of its transformation

Suppose $x$ is a non-negative random variable with CDF denoted $F$ and let $v = x+e^{-(x+1-\alpha)}$ for some $\alpha\geq 0$. Finally, let $G(v)=1-e^{-(v-\alpha)}$. Is there a $F$ and $\alpha$ such that the CDF of $v$ is equal to…
Bastiani
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Can you invert the cdf of a 2d PDF?

To generate a sample from a one dimensional PDF, I know that you can invert the cdf (assuming it's invertible) and use a uniformly distributed random number plugged into that function to generate a number from the original PDF. Is it possible to…
Alan Wolfe
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Posterior distribution of an 'exponentiated' version of a Dirichlet-categorical model

Maximum a posteriori estimates can be approached by sampling from an 'exponentiated' version of the posterior. That is, samples from $p_\eta(\theta \mid x) \propto p(\theta \mid x)^\eta$ approach the MAP estimate as $\eta\to\infty$. I was wondering…
bas
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