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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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Is there a uniform distribution over the real line?

For every interval $[a,b]$, there exists a uniform probability density over this interval, which is the constant function $f(x)=\frac{1}{|a-b|}$ for $a < x < b$, and $f(x)=0$ for all other $x$. Now it's clear that there is no ordinary probability…
Benno
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23
votes
4 answers

Independence between a constant random variable and another random variable.

Intuitively, I understand that if $Y$ is a constant random variable and $X$ is another random variable, then $X$ and $Y$ are independent. However, I can't make a formal proof because I can't show that their joint density function are the product of…
HeMan
  • 3,119
22
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4 answers

PDF of product of variables

Could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? My particular need is the following: Let $w :=u \cdot v$. The PDF of $u$…
Sebastian
  • 335
20
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1 answer

Sum of Independent Folded-Normal distributions

Let $X$ and $Y$ be independent, normally distributed random variables. How is $|X| + |Y|$ distributed? Is it known to be $|Z|$, where $Z$ is distributed normally?
Mark
  • 5,696
19
votes
1 answer

Exponential family representation of multi-variate Gaussians

I'm a bit stumped by the exponential family representation of a multi-variate Gaussian distribution. Basically, the exponential form is a generic form for a large class of probability distributions. The standard form is $$f_X(x) = \exp[\theta' T(x)…
RandomGuy
  • 530
18
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2 answers

Proof that the sum of two Gaussian variables is another Gaussian

The sum of two Gaussian variables is another Gaussian. It seems natural, but I could not find a proof using Google. What's a short way to prove this? Thanks! Edit: Provided the two variables are independent.
Nicolas Raoul
  • 325
17
votes
2 answers

Conceptual difference between Poisson and uniform distribution

I feel very stupid asking this question, because they're obviously different concepts, but I can't understand why. Every textbook I read has them both thoroughly explained, but at some point I can't grasp WHY they're different. Where do they…
FDrico
  • 199
17
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2 answers

Sum of two uniform random variables

I am calculating the sum of two uniform random variables $X$ and $Y$, so that the sum is $X+Y = Z$. Since the two are independent, their densities are $f_X(x)=f_Y(x)=1$ if $0\leq x\leq1$ and $0$ otherwise. The density of the sum becomes…
Vaolter
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4 answers

What does an empirical distribution represent?

This might sound too general, but I have a problem understanding what is an empirical distribution supposed to mean. If I take its formal definition: $P_n(A):=\frac 1 n \sum_{i = 1} ^ n I(X_i \in A)$ , how do I interpret it? Has anyone got a…
seigna
  • 317
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4 answers

What is the intuition behind the exponential distribution?

My textbook gives the definition of the exponential distribution: $$f(x) = \lambda e^{- \lambda x}$$ But I can't find a good explanation online about how this was derived/where it comes from, or the intuition behind it.
larrysummersstatistics
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15
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Is the family of exponential distributions closed under scaling?

While reading wikipedia article on Exponential distribution, I found the statement on scaling the random variable. Let $Exp(\lambda)$ be the distribution of the exponential random variable with paramter $\lambda > 0$ whose probability density…
Federico Magallanez
  • 1,878
15
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3 answers

Is There a Continuous Analogue of the Hypergeometric Distribution?

As the title states, is there a continuous analogue of a Hypergeometric distribution? If $ X \sim H(m,n,N)$ is a common Hypergeometric distribution, where $N$ is the population size, $n$ is the number of draws, and $m$ is the number of success. In…
Simone
  • 291
15
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3 answers

Proof that Conditional of Poisson distribution is Binomial

The classic example... $X \sim Po\left (\lambda\right ), Y \sim Po\left (\mu\right)$, X and Y are independent. Show that the conditional distribution of X is binomially distributed. Or in other words, $P(X=k\mid X+Y = n) = P (\tilde{X} = k),…
Jon Gan
  • 1,511
14
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1 answer

How do I find the marginal probability density function of 2 continuous random variables?

Ok, I've been looking at this problem: Let $$f(x,y) = \frac{3}{16}xy^2, \quad 0 \le x \le 2, \quad 0 \le y \le 2$$ be the joint probability density function(pdf) of $X$ and $Y$. Find $f_1(x)$ and $f_2(y)$, the marginal pdfs. Then it asks if…
EthanLWillis
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13
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If $X,Y,Z$ are iid unif[0,1], then $(XY)^Z \sim \text{unif}[0,1]$.

Here's a mind-blowing fact (to me at least) that is perhaps not so well-known: If $X, Y, Z$ are iid uniformly distributed in $[0,1]$, then $W = (XY)^Z$ is also uniformly distributed in $[0,1]$. If you don't believe me, you can check this by…
Yufei Zhao
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