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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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Could some give an examples of "a set of distributions indexed by a parameter"?

This post says The log-likelihood is, as the term suggests, the natural logarithm of the likelihood. In turn, given a sample and a parametric family of distributions (i.e., a set of distributions indexed by a parameter) that could have generated…
JJJohn
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expectation of squared summation

How does the expectation of this: $$E\Bigg[ \bigg(\sum_{n=1}^{N} x_n \bigg)^2\Bigg]$$ Equal this: $$E\Bigg[ \bigg(\sum_{n=1}^{N} x_n \bigg)^2\Bigg]=N(N\mu^2+\sigma^2)$$ I'm also told that x is Gaussian and i.i.d: $$E[x] = \mu$$ $$E[x_n^2] = \mu^2…
pico
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How to sample from unknown distribution

Let's just assume on the picture we have a prob. distribution of aliens height on a Mars. We know for sure, the area below the curve is 1. But I have a different question. How do we approach sampling from unique distributions as this is? Taking…
filtertips
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Geometric Distribution Rules

I am hoping someone could please clarify the rules for the Geometric Distribution. I know that if $X$~$Geo(p)$ : $P(X=r) = p * (1-p)^{r-1}$ $P(X < a) = (1-p)^a$ What are the rules for $P(X \ge x), P(X > x)$ and $P(X \le X)$ ?
H. Quader
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how to solve a histogram which just has the value x and no vertical or horizontal values

enter image description here The following histogram only has the values of X without a horizontal and vertical axis. Please look at the image and help me answer the questions.
Arsh Ajmera
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negative parameters in a beta distribution

I have a set of observations of credit loss data, where the mean is 37% and variance 25%. Now, I have to find the distribution and the base assumption is it will follow a beta distribution. the issue is that my alpha and beta derived from mean and…
Bik
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Why is the random variable in the uniform distribution a function?

I am taking an introductory probability theory course and we defined a random variable as a function $X: \Omega \to \mathbb{R}$. We defined $X$ to have uniform distribution on $[0,1]$ if $$\mathbb{P}(\{\omega \in \Omega: X(\omega)=x\})=…
Austin Shiner
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Understanding the Intuition for Geometric Random Variables

Could someone explain intuitively to me why for a geometric distributed Random Variable $X$ $$P(X \ge k) = (1-p)^{k-1}$$ and $$P(X \le k) = 1- (1-p)^{k}$$ I understand the pmf of the geometric distribution but don't completely understand why these…
Mathstatsstudent
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Question about marginals and Normal Distribution

Let us assume we have two continuous random variables X1 and X2 and the correlation coefficient rho. The first question is about the following assumptions: If we know that the marginals of each RV are a normal distribution and we also know that the…
Bogdan
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Poisson paradigm: Why is $\lambda$, the rate of occurrence of events, equal to the sum of the probabilities of all the events that occur?

My notes say the following about the Poisson paradigm: Let $A_1, A_2, \dots, A_n$ be events with $p_j = P(A_j)$, where $n$ is large, the $p_j$ are small, and the $A_j$ are independent or weakly dependent. Let $$X = \sum_{j = 1}^n I(A_j)$$ count how…
The Pointer
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An increasing probability density function?

Could anyone come up with a probability density function which is: supported on [1,∞) (or [0,∞)) increasing discrete
Yariv
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Is there an explicit probability density function for $\frac{x_1}{\sum_{i=1}^{n}x_i}$ where $x_i \sim U(0,1)$ and are independent random variables

Is there an explicit probability density function for $$\frac{x_1}{\sum_{i=1}^{n} x_i}$$ where $x_i$ are independent random variables and have a uniform probability density between 0 and 1.
Murali
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$F$ is the cumulative distribution function for a continuous random variable. What is the meaning of $F(b)−F(a)=0.20$?

$F$ is the cumulative distribution function for a continuous random variable. What is the meaning of $F(b)−F(a)=0.20$? Does it mean that $[a,b]$ is a length of $0.2$, or that $P(X=b)−P(X=a)=20$% or $P(X∈(a,b])=20$%. All of these options look…
Libor Zachoval
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One-to-one correspondence between Weibull and Gumbel parameters

$Y$ has the minimum Weibull $(\beta, \sigma)$ distribution if it has density function: $$f_Y(y) = \frac{\beta}{\sigma} \left( \frac{y}{\sigma} \right)^{\beta -1} \exp \left[ - \left( \frac{y}{\sigma} \right)^\beta \right].$$ $X$ has the minimum…
Marcos TV
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Probability that the second highest of three independent $U(0,1)$ random variables lies between $\frac{1}{3}$ and $\frac{2}{3}$.

Let $U_1, U_2, U_3$ be independent random variables that are each distributed uniformly in $(0,1)$. What is the probability that the second highest value among them lies between $\frac{1}{3}$ and $\frac{2}{3}$. Could someone help me solve this? Let…
Satish Ramanathan
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