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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Tuning Beta model: Bayes Rules! Book exercise

I'd like to verify my solution and receive wider clarification where it's possible. This is an exercise from Bayes Rules! book. Exercise 3.1 (Tune your Beta prior: Take I) In each situation below, tune a Beta(α,β ) model that accurately reflects…
Raibek
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Deriving the pdf of $Y$ given that $X$ is $N(\mu,\sigma^2)$.

Claim: If $Y=e^X$ has a lognormal distribution whereby $X$ ~ $N(\mu,\sigma^2)$, then the pdf of $Y$ is $g(y)=\frac{1}{y\sqrt{2\pi\sigma^2}}exp[-(lny-\mu)^2/2\sigma^2], 0
Karam
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Can a probability density function be used directly as probability function?

This might be something basic but it confuses me greatly. I am reading a literature, where they use the probability density function of a Gaussian distribution, that is $$f(x)=\frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2}…
Karel
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Calculate Var(6X+2) knowing that EX^2 = 6, EX = 2

As mentioned, i have to calculate $\mathrm{Var}(6x+2)$ knowing that $E(X^2) = 6$ and $EX = 2$ I know that $$\mathrm{Var}(6X+2) = 6^2\mathrm{Var}(X) = 36 \mathrm{Var}(X) = 36 \cdot ( E[X^2] - E^2[X] ),$$ but i kinda stopped at this point because i…
learningbtw
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Joint distribution of sample from urn

I'm doing a basic question that specifies there are $p$ black balls, $q$ white balls, and $r$ red balls in an urn. You draw $n$ balls without replacement. I'm looking for the joint distribution of the number of black, white and red balls in the…
fmtcs
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How to find the output for a given input following a Pareto Distribution?

Assuming a Pareto Distribution [1], it's commonly known that 20% of the effort gives 80% of the results. The "80–20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and…
valbaca
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Log normal distribution

Given a random variable $X$ with log normal distribution, can we find the probability of $X$ being greater than a positive constant $a$, i.e can we determine the integral $$ \int_a^\infty \frac{1}{xs\sqrt{2\pi}} e^{-(\ln(x)-s)^2/(2s^2)} dx $$ from…
adsj
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How to prove if these two random variables are independent or not independent?

I came across this question Let X and Y be independent random variables with distributions. Let $Z = XY$ Write down a table giving the probability distribution of $Z$ Are the random variables $X$ and $Z$ independent? So I managed I already…
Bryan Hii
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Is it true that the minimum value of kurtosis for any distribution is -2?

The Bernoulli distritbution has excess kurtosis = $-2$. In addition for a symmetric beta distribution with $\alpha = \beta$ the excess kurtossis is $-6/(2\alpha+3)$ whose minimum value is again $-2$. But, how can it be shown whether this is the…
Davius
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Conditional probability in exponential distribution

Q. The amount of time that a watch will run without having to be reset is a random variable having Exponential distribution with mean $\mu=50$ days. Find the probability that such a watch will (i) have to be reset in less than 20 days. (ii) not have…
Riaz
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Does the average of n iid random variables eventually stochastically dominate the average of n iid random variables with lower expected value?

Let $(X_1, X_2,...)$ be i.i.d. random variable with finite mean $E(X)$ and let $(Y_1,Y_2,...)$ be i.i.d. with mean $E(Y)
DM-97
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Random variable with pdf $-\log x, x\in (0, 1]$

Is there a name for the probability distribution with the following pdf? $$f(x) = \begin{cases}-\log x, & x\in (0, 1] \\ 0,& \text{otherwise}\end{cases}$$ (Observe that $\lim_{t\to 0}\int_t^1 f(x)dx = 1$.) The cdf is $$\int_0^x -\log(t) dt = 1 -…
Max
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How do I determine the exact domain of y given a graph?

I've made a post earlier about finding the joint PDF of a given figure: $$f_{X,Y}(x,y) = \begin{cases} 1\over 2& ,0\le x \le 2 , \max(0, x-1) \le y \le \max(1, x) \\ 0 & \text{,otherwise}\end{cases}$$ Someone told me how to find it using the area…
Andrei0408
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How to determine the joint PDF given a graph?

Given the following graph: $$f_{X,Y}(x,y) = \begin{cases} a &, x,y \in OABCDE \\ 0 &, \text{otherwise}\end{cases}$$ My solution: $$f_{X,Y}(x,y) = \begin{cases} a &, 0 \leq x \leq 2, 0 \leq y \leq 2 \\ 0 &, \text{otherwise}\end{cases}$$ Is this…
Andrei0408
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Probability of putting at least one colored ball in each bucket

Let's say I have 20 buckets and each bucket will have exactly 50 balls randomly inserted into them from a group of 1,000 colorless balls. Of the 1,000 balls, I can color as many balls as I want red. How many balls would I have to color red to make…
Max
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