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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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probability density function of a function of a random variable?

Say we have a random variable $X$ with some probability distribution $D$. For example, $D$ could be the normal distribution: $$\text{pdf}_X(x)=D(x)=\frac{1}{\sqrt {2\pi \sigma ^2}}\int_{-\infty}^{\infty}e^{1\frac{x^2}{2\sigma^2}}dx$$ Now let's say…
user56834
  • 12,925
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Sufficient conditions for $X/(X+Y)$ to have a uniform distribution

Suppose that $X$ and $Y$ are i.i.d. r.v.'s with an exponential distribution of parameter $1$.Then it is known that the ratio $$Z = \frac{X}{X+Y}$$ has a uniform distribution on $(0,1)$. See for instance: X,Y are independent exponentially distributed…
mlc
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Find the distribution of $\log\left( \frac{\min(U_1,U_2,\ldots, U_n)^n}{U_1 U_2 \cdots U_n} \right)$ for iid $U_k \sim U(0,1)$

I stumbled on a curious problem: Choose $n \in \mathbb{Z}_{\geqslant 2}$, and consider independent identically distribution random variables, uniformly distributed on the unit interval. What is the distribution of $$ W_n =…
Sasha
  • 70,631
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What is the probability that two univariate Gaussian random variables are equal?

Let $X_1$ and $X_2$ be two independent univariate Gaussian random variables, s.t. $$X_1\sim \mathcal N (m_1,\sigma_1^2)$$ $$X_2\sim \mathcal N (m_2,\sigma_2^2)$$ So now what is $P(X_1=X_2)$? I tried in the following…
Chehnobot
  • 63
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scores of individuals and evaluation

Suppose we have a fixed (ordered) set of $2000$ integers $p_m$ drawn from a discrete uniform distribution on $\{1,2,...,100\}$ arranged in a terrain. Let this terrain be denoted $\mathcal{T} = \{p_1,p_2,...,p_{2000}\}$. We also have $N = {\ell…
afedder
  • 2,102
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Generalized $\chi^2$ distribution?

Consider each $X_i \sim N(0,1)$. Then the random variable $Y=\sum_{i=1}^n X_i^2$ is a $\chi^2$ distribution with $n$ degree of freedom. Is there any probability distribution about a weighted sum of the square of standard normal random variables…
Endo
  • 163
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Looking for Anthony?

You are trying to locate an old high school friend who lives in Chicago. Unfortunately, your friend's name is Anthony Smith and the Chicago phone book lists phone numbers for $24$ different people named Anthony Smith. (Assume that your friend's…
Mark Dav
  • 51
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Legitimacy of a Bivariate Distribution Function

I am asked to explain why the following function is not a legitimate (multivariate) distribution function. $$ F(x,y) = 1 - e^{-x-y}, x,y \geq 0 $$ I am tempted to reason as follows: The function has right continuity on each of its variables, so its…
user191919
  • 713
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Is there a general formula for the mixture of two beta distributions?

I'm wondering if there's a general formula to simplify the mixture of two beta distributions. Ex. I have $.5\operatorname{beta}(a_1,b_1) + .5\operatorname{beta}(a_2,b_2)$. Can I find $a_3$ and $b_3$ such that this mix is equivalent to a…
Pburg
  • 1,226
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Concept of Probability distribution

Sorry for a silly question, but it seems like only you can answer this question. What's is the concept of Probability distribution, what's the meanining behind this term. Why we need probability function if we already have pdf (probability density…
com
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Find marginal density function from joint density function

If I have a joint density function for X and Y: $f_{X,Y}(x,y) = \begin{cases} \pi x \cos(\frac {\pi y} 2) & 0 \le x \le 1, 0 \le y \le 1 \\ 0 & \text{otherwise} \\ \end{cases}$ How do I find the marginal density function for X? I think I need to…
Jigglypuff
  • 153
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What is the distribution of the difference of two normalized binomial random variables?

Let $X \sim Bin(n, p)$ and $Y \sim Bin(m, p)$. How is $$Z_1 = \frac{X}{n} - \frac{Y}{m}$$ and $$Z_2 = \left|\frac{X}{n} - \frac{Y}{m}\right|$$ distributed? (Hence: What is their cumulative distribution function?) Background I am mainly interested in…
Martin Thoma
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Why does $\frac{X}{X + Y} \sim\mathrm{ Beta}(\frac{n}{2}, \frac{m}{2})$?

If random variables $X \sim \mathrm{Gamma}(\frac{n}{2}, \frac{1}{2})$ and $Y \sim \mathrm{Gamma}(\frac{m}{2}, \frac{1}{2})$, where $m$ and $n$ are constants, why does $\frac{X}{X + Y} \sim \mathrm{Beta}(\frac{n}{2}, \frac{m}{2})$? In general, can we…
David Faux
  • 3,425
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3 answers

Calculation of inverse of chi-square's expectation

I don't know where to begin to calculate the expectation value of the random variable $1/V$, where $V$ is a random variable with chi-square distribution $\chi^2(\nu)$. Could somebody help me?
kevincuadros
  • 230
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Derivation of the Logistic distribution

The logistic distribution is well known. For example, the standard pdf of the logistic distribution is given as: $$ f_X(x) = \frac{e^x}{(1+e^x)^2},\,\,-\infty\lt x\lt \infty~~~~~~~~~~(1)$$ My question is this: How did this distribution come about?…
Gorg
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