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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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How do you calculate the expected value of geometric distribution without diffrentiation?

Is there any way I can calculate the expected value of geometric distribution without diffrentiation? All other ways I saw here have diffrentiation in them. Thanks in advance!
user21312
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Is the y axis on a PDF actually meaningless?

This idea popped in my head when I was reading this post on the normal distribution and the y-axis. My question is (and taking advantage of a nearby computer), a PDF inputs one value and returns another, and this returned value is a probability.…
tumultous_rooster
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Finding the probability density function of $Y=e^X$, where $X$ is standard normal

Let the random variable $X$ have the $N(0,1)$ distribution for which the probability function is: $$ f(x)= \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right), -\infty< x <\infty $$ Let $Y=e^X$. A. Find the probability density function for $Y$, B.…
Sandrochka
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Why does Pearson's chi-squared test divide by the mean and not the variance?

I am wondering why in Pearson's chi-squared test, the divisor of each element in the sum is the matching expectation and not the matching variance. As I understand it, the test works by standardizing each normal variable before summing, so the…
Danra
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Does the Poisson distribution work for non-integer exponents?

The question regards the Poisson distribution function as given by: $$\frac{x^k e^{-x}}{k!}$$ The distribution's domain (x) goes from 0 to $\infty$, and $k \in \mathbb{N_0}$ I tried the distribution as the following function: $$\frac{x^r…
user132716
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2 answers

Adjusting poisson distribution lambda

If I have a poisson distribution for some variable x such that $\lambda = 100$, is it possible to linearly adjust that variable to generate probabilities for smaller increments? For example, if I have on average 100 customers arriving an hour…
Zxaos
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Whats the formula for the probability density function of skewed normal distribution

The formula for the probability density function of a standard normal distribution that isn't skewed is: $$P(x) = \frac{1}{\sqrt{2π}}e^{-(x^2 / 2)}$$ where, $π = 3.14, e = 2.718$. What if it is skewed left or right like this: What is the formula…
Hauzron
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Why does marginalization of a joint probability distribution use sums?

I'm going through a book that talks about probability distributions. The part which is tripping me up is conceptual. It says: "We can recover the probability distribution of any single variable from a joint distribution by summing (discrete case) or…
Capstone
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compound of gamma and exponential distribution

What is the distribution of a exponential distribution, whose parameter is drawn form the gamma distribution $$ X \sim \operatorname{Gamma}(\alpha,\beta)$$ $$ Y \sim \operatorname{Exp}(X)$$ how is $Y$ distributed? EDIT: Since there are different…
dylan2106
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If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed?

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed? Any explanation would be very appreciated.
Freeman
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Cumulative distribution function of sum of binomial random variables

I was wondering how to get the cumulative distribution function of a sum of two random binomial variables. X + Y, where X has n=15 trials and Y has m=15 trials and probability=0.2 for both P(15<= X+Y <= 20) how would I express this as a cumulative…
Alistair
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Are there any (pairs of) simple distributions that give rise to a power law ratio?

If I recall correctly, for $X$, $Y$ normally distributed, the ratio $X/Y$ is Cauchy-distributed. This is sort of like a power law, but isn't quite. So: Are there any simple distributions for two RVs $X$, $Y$ s.t the ratio is really power-law…
S Huntsman
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moment-generating function of the chi-square distribution

How do we find the moment-generating function of the chi-square distribution? I really couldn't figure it out. The integral is $$E[e^{tX}]=\frac{1}{2^{r/2}\Gamma(r/2)}\int_0^\infty x^{(r-2)/2}e^{-x/2}e^{tx}dx.$$ I'm going over it for a while but…
monte13
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Is the sum of independent unimodal random variables still unimodal?

Is the sum of independent unimodal random variables still unimodal? If yes, can you please give me some hint on why this holds? If no, can you show me some counter-example and suggest under what condition the sum remains unimodal? Thank you in…
sinoTrinity
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Mean And Variance Of Beta Distributions

I want to find mean and variances of beta distribution . The distributions function is as follows: when $x$ is between $0$ and $1$ $$ f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}} \, du $$ Searching…
Way to infinity
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