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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Conditional distribution of multivariate normal

$X \sim N(\mu, \Sigma)$. How can I find conditional distribution of $X$ given $AX$, where $A$ is a non-random matrix? I know that $AX \sim N(A\mu, A\Sigma A^T)$ but don't know what to do next. P.S - I am preparing for an exam and solving problems…
Markoff Chainz
  • 309
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1 answer

Distribution function of X given a table of probability function

I am shown a table with x and f(x) where f is the probability function of a random variable X. x | 1 | 2 | 3 | ----------------------------- f(x) | 1/2 | 1/4 | 1/4 | What would I need to do to find the distribution function of X…
rhino18
  • 23
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Distribution of ratio of sums of gamma distributions

If $X_1,X_2$ are independent r.v.s with $X_1 \sim \Gamma(\alpha,\theta)$, $X_2 \sim \Gamma(\beta,\theta)$ then it is known that $$\frac{X_1}{X_1+X_2} \sim \text{Beta}(\alpha,\beta)$$ Let $X_i$ be iid with $X_i \sim \Gamma(\alpha,\theta)$. What is…
Haderlump
  • 375
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What is the correct way to combine disjoint probability density value?

I'm building a particle filter and have two measurements, say one from a camera (C) and one from a GPS (G). Each measurement has a PDF associated with it defined by a mean and sigma. It turns out that the sigma for C is very small while the one for…
CPayne
  • 121
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2 answers

minimize variance

$X_1$ and $X_2$ are independently distributed random variables with $$P(X_1=\Theta+1) = P(X_1=\Theta-1) = 1/2 \\ P(X_2=\Theta-2) = P(X_2=\Theta+2) = 1/2$$ Find the values of a and b which minimize the variance of $Y=aX_1 + bX_2$ subject to the…
dru
  • 21
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Probability distribution of order statistics

Let $X_1$, $X_2$ and $X_3$ be independent random variable with continuous distribution $$f(x;\theta)=\frac{1}{\theta}I_{(0,\theta]}(x), \ \theta \gt 0$$ I need to find distribution of $Z=\frac{X_{(3)}}{\theta}$, where…
Paul
  • 529
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Problem with the density of the compound distribution

My problem is to calculate $E[\max(S-5000, 0)]$ where $$S = \sum_{i=1}^{N} X_i,$$ $N$ is a random variable with geometric distribution, parametrized as follows: $$P(N=n) = \frac{\beta^n}{(1+\beta)^{n+1}}, ~~~~~ n=\{0,1,2, ...\}$$ and ${X_1,…
Jerry
  • 31
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continuous variable $X$ such that $\Pr(X=x) \neq 0$, for some value $x$.

I'm interested in cases where you have a probability function $\Pr$ defined over the values of the continuous real valued variable $X$, where some particular value or values have non-zero, non-unitary probability, and the remaining probability is…
JDG22
  • 81
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What is the difference between infinitely divisible and stable law?

I read somewhere that stable law is the special case of infinitely divisible. In other word, stable distribution is a special case of infinitely divisible distribution. But I am not quite sure what differentiate both of them apart.
Liam Nousa
  • 211
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How to continue on this Continuous Probability density function?

Let $f(x)$ be a continuous probability density function. Show that, for every $-\infty < µ < \infty$ and $σ > 0$, the function $\frac{1}{\sigma}f(\frac{x-\mu}{\sigma})$ is also a probability density function. Well I know that if $f(x)$ is a cont.…
llawliet_78
  • 323
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1 answer

Integrating a Multinomial distribution over a Dirichlet distribution

I need to compute the probability of sampling a specific vector from any multinomial generated by a Dirichlet parametrization. Let's call the vector $\mathbf{x} = \{x_i\}, i \in K$ I need to compute $$ p(\mathbf{x}|\alpha) = \int Mult_{PMF}(x, y) \,…
Makers_F
  • 143
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1 answer

Mod of a random variable

I had this problem where I wanted to generate random variables (discrete) in a way that certain numbers were more probable than others (basically geometric) but since I wanted to use this number as an array index, I wanted it to be bounded between…
Wajahat
  • 225
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What is joint characteristic function of filtered version of gaussian random variable

Problem: Let $U_{n}$ be the discrete-time iid Gaussian random process with mean 0 and variance 1. Two filtered version of $U_{n}$ is defined by $X_{n}=U_{n}+U_{n-1 }$ and $Y_{n} = U_{n} - U_{n-1}$. Find the joint characteristic function when…
Kim
  • 179
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3 answers

probability density function chi squared

Exercise I've been tasked with deriving the probability density function for a chi-squared random variable $$f(x;q) = \begin{cases} \hfill 0 \hfill & x\leq 0 \\ \hfill…
Blake
  • 2,610
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1 answer

Show that $g(x,y) = \frac{f(x+y)}{x+y}$ is a density function

Problem: Let $f$ be the pdf of a positive rv and write $g(x,y) = \frac{f(x+y)}{x+y}$, if $x>0,y>0$. Show that $g$ is a density function in the plane. $g$ is a pdf if $\int_{0}^{\infty}\int_{0}^{\infty}g(x,y)dxdy = 1$, or equivalently if…
sim
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