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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Expected value and variance of discrete random variable

Let $Y$ be a discrete random variable with density function: $$p(y;\theta)=\left(\frac{\theta}{2}\right)^{\lvert y\rvert}(1-\theta)^{1-\lvert y\rvert}$$ where $y\in\{-1,0,1\}$ and $\theta \in[0,1]$. I have to find the expected value…
Paul
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Indicator variable syntax

Are these syntax equivalent? $$f(x,\lambda)=\lambda e^{-\lambda x}I_{(0,+\infty)}(x),\ \lambda > 0$$ $$f(x,\lambda) = \left \{ \begin{array}{cl} \lambda e^{-\lambda x} & x \gt 0 \\ 0 & \text{Otherwise} \end{array} \right .$$
Paul
  • 529
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Simulating Probability Distributions

We have the following cumulative distribution function: $$ F(x) = \begin{cases} 0 & x < 0 \\ x^2/9 & 0 \le x \le 3 \\ 1 & x > 3 \end{cases} $$ To find $X$ in terms of $U \sim \mathrm{Uniform}[0, 1]$: $$ F^{-1}(u)=\min\{x: F(x) \ge u\}=\min\{x:…
user137481
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Probability Distribution of Random Variable

If our random variable only has six equal possible outcomes, will any probability distribution resulting in mapping to real numbers consist of only six real numbers each with probability $\frac{1}{6}$ and the rest of the real numbers with…
SwabianOrtolan
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Find constants so that a random variable has a chi-squared distribution

Suppose $X$ ~ $N(3,5)$ and $Y$ ~ $N(-7,2)$ be independent. Find constants to satisfy: $\quad C_1(X+C_2)^2$ + $C_3(Y+C_4)^2$ ~ $\chi^2(C_5)$ I started with: $\quad Z_1=\sqrt{C_1}X+\sqrt{C_1}C_2$ ~ $N(3\sqrt{C_1}+C_2\sqrt{C_1}, \;5C_1)$ $\quad…
user137481
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drop points in a line which obey uniform distribution

There is a line which can be considered as an interval [0,L], here we drop N points randomly on the line which obey the uniform distribution, namely, the probability of the location of any points in the line is equal. Now the distance between any…
user330187
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Show that $R^{2}$ and $\theta$ are independent and $R^{2}\sim U(0,1)$, $\theta\sim U(0,2\pi)$ in marsaglia's method

I have little problem to show that $R^{2}$ and $\theta$ are independent in marsaglia's method and furthermore $R^{2}\sim U(0,1)$ and $\theta\sim U(0,2\pi)$. For the first method (Box & Muller) take two random variable $U_{1}$ and $U_{2}$ such that…
user123456789
  • 165
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Finding the distribution of a function of random variables using the definition (without the convolution theorem)

I'm trying to find $f_Z(z)$ with $Z=2X-Y$, for $X$ and $Y$ with joint density function $f_{XY}(x,y)$: $$ \begin{cases} x/8 & 1 \le x \le 3 \land -1 \le y \le 1 \\ 0 & \text{elsewhere} \end{cases} $$ and I want to do it without using the convolution…
totota
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PMF $Pr(X = x) = 0.8 \cdot 0.2^{x-1} \space,\space x\geq 1$ Question!

What am I given? In an AI model of a tennis simulation video game, it is assumed that the number of shots until someone wins a point, X, has a geometric distribution with probability mass function: $$Pr(X = x) = 0.8 \cdot 0.2^{x-1} \space,\space…
Rubicon
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Rolling a die that follows a poisson distribution and computing it's mean

Roll a fair, 4-sided die N times where N is a Poisson random variable with parameter λ>0, let X be the number of 3's rolled in this experiment. Find E(x) What I have figured out is that E(X) seems to be N/4. But am doubting this since we have been…
Ben
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Inverse probability distribution limit

I have a very simple experience $E$ that takes a time $T$ to complete, $T$ is uniformly distributed in $[1;2]$. I consider doing a sequence of such experiences $E_i$ ($i\le n$). By Central Limit Theorem, the total time $T_n=\sum_{i\le n} T_i$ is…
Xoff
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Distribution of the sum of two (dependent?) random variables

There are two random variables $X$ and $Y$, each of which can take on the values $0$ or $1$. Furthermore: $P(X=0,Y=0)=p$, $P(X=0)=1/2$ and $P(Y=0)=1/2$. So these two shouldn't be independent in general, since $\frac{1}{2}^2$ is not necessarily…
355durch113
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Intuitive explanation of distribution of ratio of independent random variables

Case 1: I have two independent exponentially distributed random variables $X$ and $Y$. Intuitively, it makes sense that the sum of those variables is essentially exponentially distributed, but is that correct? Case 2: I have a uniform random…
Nikos A.
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Any one can help finding a distribution function?

I am looking for a distribution function with the following characteristics: F_x(x) is Continuous, differentiable and truncated on some [a,b]. I also need that that the density around the lower bound will be of measure zero, meaning: f_x(a)=0.
MoRkO
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How can i compute the probability that a cloud of points was created by a probability distribution?

I have a number of probability distributions that describe a number of points. like this: Now if i have draw a one point out of each distribution, i get a bunch of points randomly set on the 2-dimensional surface. How can i, given the set of…
tarrasch
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