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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Variance of Categorical Distribution?

I know this sounds like something I can just Google, but surprisingly, I cannot find it ANYWHERE. All I found is a document that says: "We will skip the variance of categorical distribution here." Source:…
A_for_ Abacus
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Joint distribution of two non-independent standard normal distribution random variables.

We have ${x_1, x_2}$ ~ ${N(0,1)}$ and it is not said that they are independent. Show that then ${(x_1, x_2)}$ ~ ${N(a,b)}$ , where a - vector of mean values and b is covariance matrix. I know that if ${x_1, x_2}$ independent we can get bivariate…
klani
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Prove this property of characteristic function

I have been trying to prove this for some time now but I have reached nowhere. $\phi(X)$ is the characteristic function of rv X. Prove: $\phi(X+Y) = \phi(X)\phi(Y)$ if X and Y are independent rv. I tried expanding them as E[$e^{itX}$] but that…
Alex
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Approximation of binomial coefficient using Stirlings formula

We have the approximation: $\ln x! \simeq x \ln x - x + \tfrac{1}{2} \ln (2 \pi x)$. I want to estimate $\ln \binom{N}{r} = \ln\left(\frac{n!}{(N - r)! r!}\right) = \ln(n!) - \ln((N - r)!) - \ln(r!)$. The result I want to get is $$(N - r) \ln(N / (N…
user581023
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What is the meaning of mean of a probability distribution table?

Suppose there is a random variable X which takes the values as $X = {0, 1, 2, 3, 4}$. Each number in this set denotes the total number of wins (head) in a coin game where an unbiased coin is tossed $4$ times. Then $$P(X_i) = (\frac{1}{2})^4,…
Aman Jain
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How can I use Joint Distribution Function to Find the Probability for 2 random variables in a certain interval?

We are given this density function : $$ f(x,y) = \begin{cases} \frac{xy}{96}, & 0
g0x0
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Supporting set of $Y = M\left( {1 - \frac{{1 + N}}{{{e^{a \times P \times X}} + N}}} \right)$ where $X$ is exponentially distributed?

Given the random variable $Y = M\left( {1 - \frac{{1 + N}}{{{e^{a \times P \times X}} + N}}} \right)$ with the following real number $N>1, a>0, P>0, M>0$ and $X$ is an exponential random variable with the corresponding PDF: ${f_X}\left( x \right) =…
Tuong Nguyen Minh
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La distribution vp(1/x)

Pour $\phi \in \mathcal{D}(\mathbb{R})$ tel que $Supp(\phi) \subset [-R,R]$, pourquoi $\int_{|x| \geq \epsilon} \frac{\phi(x)}{x}dx = \int_{R \geq |x| \geq \epsilon} \frac{\phi(x) - \phi(0)}{x}dx$ ?
Romain44
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Conjugate prior Gamma distribution on Poisson intensity

I don't know what I am missing in following in my understanding. Whether it is my mathematica code that is incorrect or my mathematical skill is short. Gamma distribution is the conjugate prior when the likelihood function is Poisson distribution.…
kaka
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Construct a sequence.

We know that, if a sequence $\{\xi_n\}_{n=1}^{\infty}\subset [0,1)$ is equidistributed, then it must be dense in $[0,1)$. My problem is, how to construct a sequence $\{\xi_n\}$ that is dense but not equidistributed?
ALe0
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CDFs of generalize beta distribution pdf and standard beta pdf.

Let $f(x)$ be the probability density function (pdf) of the standard beta distribution on $(0,1)$. And let $f_d(x)$ be the pdf of the generalized beta distribution on $(0,d)$. I know that, $$f_d(x) = d \cdot f(\frac{x}{d})$$ The cumulative…
Legendre
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Distribution rolling two dice

Let's suppose we are considering the experiment of rolling two dice and X is the random variable that tracks the number of one die and Y the number of the other die. --> What are the distributions of X (and Y) ?
Lea67
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Where did I make a mistake calculating the distribution of the sum of two independent exponentially distributed random variables?

We know that if both $X$ and $Y$ are continuously distributed with PDFs $f^X$ and $f^Y$, then $X+Y$ is continuously distributed as well with density function $f^{X+Y}(x) = \int f^X(z)f^Y(x-z)\text{ d}\lambda(z)$ where $\lambda$ denotes the Lebesgue…
Nicolas
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Derive the distribution of $V=-\beta\sum_{i=1}^\alpha\log_e{F\left(Y_i\right)}$

This was a part question to what I posted here and it went like this: Suppose $Y_1,Y_2,...$ are independently and identically distributed random variables with common distribution function $F$. For any positive integers $\alpha$ and $\beta$, derive…
DeBARtha
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Density Function of $F\left(Y_1\right)+F\left(Y_2\right)$

I came across a question which stated: Suppose $Y_1,Y_2,...$ are independently and identically distributed random variables with common distribution function $F$. Then find the pdf of $$U=F\left(Y_1\right)+F\left(Y_2\right)$$ From what I…
DeBARtha
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