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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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Why does $\min(X_1, X_2 ,\ldots, X_n) \sim \mathrm{Expo}(X_1 + X_2 + \cdots + X_n)$?

Let $X_1, X_2,\ldots,X_n$ be identical and independent random variables that are distributed exponentially with rate value $\lambda$. Then, does $\min(X_1, X_2 ,\ldots, X_n) \sim \mathrm{Expo}(X_1 + X_2 + \cdots X_n)$? I think I have a proof: Let…
David Faux
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Probability of X or more simultaneous/overlapping events at any point over time period Y given Z agents issuing N events/second of average duration M?

This might be a "please do my homework" question, except I'm a humble developer trying to predict system behavior rather than a student... I have N devices issuing potentially-expensive http requests - each request takes, say, 2 seconds to complete…
Tao
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Estimate the density function of a distribution based on binomial distributions.

Let's consider a set of nodes $V$, and let some nodes be colored with one color choosen between two possible colors; denote the color $\alpha$ and $\beta$, with respectively $I>0$ and $K>0$ nodes colored with them. Note that we could have $K+I
Immanuel Weihnachten
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Can one decompose a random variable $X$ into a mixture of $Y$ and $Z$ with prespecified mixing probabilty $p$ subject to the constraint $E[Y] = \mu$?

Let $X$ be a random variable with known distribution function $F$ and let $Q \sim Bernoulli(p)$. Conditional on $Q=1$, $X = Y$ where $Y \sim G$. Conditional on $Q = 0$, $X = Z$ where $Z \sim H$. Now suppose we fix a value for $p$, strictly between…
inhuretnakht
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Using Binomial Distribution to evaluate this probability distribution

I am trying to find the value of a skewed distribution but can't make sense of what to plug in to evaluate the answer. This is the given: $$ \text{Let X be Binomial(n, p). } \text{Using that, evaluate:} $$ $$ \beta =…
Kay
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Probability density with no local max?

Let $f$ be a probability density function that is continuous and positive everywhere on $\mathbb{R}^n$, $n>1$. Is it possible that $f$ has no local maximum? For $n=1$ it is easy to show that this can't happen. But the bivariate case seems…
BobS
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What is the pdf of $Y = \log X$?

Let $X$ be a standard exponential random variable, and $Y = \log X$. (a) Find DIRECTLY the c.d.f of $Y$ and use it to calculate the density of $Y$. (b) Find DIRECTLY the p.d.f of $Y$. So far, I did: (a) $F_Y(y) = P(\log X \leq y) = P(X \leq e^y) =…
woaini
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How can I calculate how many times I need to upgrade in this game?

A bit of an oddball question. I'm playing a game where I have a certain number of items, $n$ (in this case 9), and a certain action upgrades each item by one with a chance of $\rho$ (in this case, 33%). I'm trying to get all of the items to a…
user66698
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Is the property of the mean being equal to the variance unique to the Poisson distribution?

Just asking out of curiosity. A homework question asked for the mean and variance of a distribution, and they turned out to be equal. It's not obvious to me that the distribution in question is poisson though.
Reylined
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Probability distribution of $e^X$, when $X ~$ uniform$(0,1)$

What is the probability distribution of $e^X$, when $X$ is a random variable and follows the uniform distribution $U(0,1)$? I noticed that the probability of $e^X$ decays exponentially from $1$ to $e$, even though probability distribution of…
Tony
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Is the set of probability distributions infinite in measure?

I've been learning about more probability distributions that are not covered in introductory statistics, and I realized looking at Wikipedia that there are a lot that have been named. This got me wondering, is there a countably infinite number of…
Galen
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Exponential distribution from Poisson

In Poisson distribution, the probability of inter arrival time to be t or less is: $$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$ and probability of one arrival in t is: $$ P(k=1)= \lambda t e^{- \lambda t} $$ I…
Betamoo
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Distinct items in a Sample of a Zipfian Distribution

The Zipfian distribution serves as a good model for several interesting things. For example, the rate of occurrence of words in the English language (or most any language) appear to follow a Zipfian distribution. Let's say I have a Zipfian…
John Berryman
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Show that $\mathbb{E}(X-c)^2$ is minimum when $c = \mathbb{E}(X)$

Suppose that the random variable X has the cumulative density function F(x). Show that the expected value of the random variable $(X-c)^2$ is minimum if c equals the expected value of X. I know that the cumulative distribution function ("c.d.f.") of…
Mestica
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Composition of Poisson arrivals and Bernoulli RVs

I was wondering if there is a known characterization of an arrival process defined as follows: a "potential" arrival occurs according to a Poisson process of rate L, then a Bernoulli RV with P(1)=B determines whether the arrival officially occurs. …
Aaron Kolb
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