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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What is the expected area of this enclosure?

I have no clue how to do this textbook question: An environmental artist is planning to construct a rectangle with 36m of fencing as part of an outdoor installation. If the length of the rectangle is a randomly chosen integral number of $L$ metres,…
Noah Woodworth
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Probability distribution of $Z = F(X)$ if that of $X$ is known

Let $X$ be a random variable with probability distribution function: $F(x) = \begin{cases} 1 - ae^{-x/5} : x \ge 0 \\ 0 : x < 0 \end{cases} $ Let $ Z = F(X)$, find the probability distribution function of $Z$. My attempt: $F(z) =…
dunguyen
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Average number of trials until we get $1$ success and $1$ failure

The probability of a success is $0.81$. The probability of a failure is $0.19$. What's the average number of trials until we get both outcomes (a success and a failure)? I started with E[X] = 2*2(0.81 * 0.19) + 3(0.81 * 0.81 * 0.19) + 3*(0.19 *…
user6952624
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Maclaurin series expansion of the mgf of the standard normal distribution

We were asked to use the Maclaurin series expansion of the moment generating function of the standard normal distribution. Please explain why the rth moment about the mean is 0 when r is odd. Thank you
ANONYMOUS
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the binomial distribution form to determine the possibility of rolling exactly five 3’s.

Let’s say that a die is rolled seven times. Use the binomial distribution form to determine the possibility of rolling exactly five 3’s. Please help me! I'm so confused. BTW i'm really bad at math so please explain in a really simple way. Thanks!
Maria
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How to find cumulative distribution function given the first ocurence probability?

Let the function $a(t),~ t \ge 0$, denotes the probability that there is an arrival in the current period (period $t$) given that no customers arrived in the previous $t-1$ periods. Assuming the inter arrival times to be independent and identically…
Litun
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Gamma distribution from the sum of independent exponential distributions

From a paper I'm currently reading: In the simplest setting, the average waiting time (or equivalently the departure rate) in each stage is assumed to be equal: the overall infectious period is then described by the sum of n independent exponential…
Fomite
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discrete uniform distribution for selecting at least two same numbers

$M$ integers are drawn, each randomly and independently from a discrete uniform distribution with range between $1$ and $K$ (both included). So each of these $M$ random numbers will be an integer between $1$ and $K$ (both included). What is the…
anonymous
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Closed form for sum of fraction of binomials

I (think I) solved the problem of finding the distribution of random variable $Y=k$ denoting the largest out of $m\leq n$ balls, drwan out of an urn containing all balls labled $1,\dots,n$, without replacement. Using combinatorics I'm quite sure…
gbi1977
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Odds of placing top 20 in a contest of 200 participants out of 5 contests with the first score being 193

I am in a contest with 200 total participants. There will be 5 total scores to determine the top 20 finishers. Each is initially ranked according to their ranking from a previous contest. That previous ranking is your first score. Mine is 193. The…
S. Tone
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$X_1, X_2, \ldots$ follow identical, independent $\operatorname{Ber}(p)$ distribution. $N = \min\{ n \ge1: S_n = 1\}$

$X_1, X_2, \ldots \sim \operatorname{Ber}(p)$, those variables are i.i.d. Suppose $S_n = X_1 + X_2 + \cdots + X_n$ $$N = \min\{ n \geq 1: S_n = 1\}$$ 1) Show that $X_{N+1}, X_{N+2}$ also follow i.i.d $\operatorname{Ber}(p)$; 2) show that $\{X_1,…
Olympia111
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Why is the value of the probability function grater than $1$?

Can you explain me, why $P(x)>1$ or why is the value of the function $≈1.2$ ? $P(x) ≤1$, right?
user548054
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Why doesn't this work for finding the PDF of $X_1 X_2$ where $X_1,X_2$ iid Unif(0,1)

Let $X_1,X_2$ be iid Unif(0,1) random variables. Find the PDF of $Y = X_1 X_2$. This is my method and I'm unsure why it's wrong: $$\mathbb{P}(Y\leq y) = \mathbb{P}(X_1 X_2 \leq y) \\ = \int_{\text{all} x} \mathbb{P}(X_2 \leq y/x | X_1 =…
Mr. Bromwich I
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How to find the individual probability density functions given a joint probability density function

I came across this question. Essentially, the answer says that if you can write a joint pdf in the form $p(x,y)=p_1(x)p_2(y)$, then the random variables are independent of each other. My question is this: Given a pdf: $$p(x,y)=4e^{-2(x+y)}$$ How…
quanticbolt
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Sum of 4 random variables such that their sum is 1 and each is U(0,1)

I'm trying to formulate a way to generate a random set of 4 numbers that sum to 1 (a discrete probability distribution) such that each is uniform, ~$U(0,1)$. They obviously can't be iid, and normalized uniforms have a mean 0.25 which is way too low…
jrclimer
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