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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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Monotonic Polynomials

I need to fit a cumulative distribution that is monotonically decreasing. I have in mind to use a polynomial of the lowest order that will give a fit that is looks good to the eye. Can someone suggest a polynomial functional form that is…
jhadams1
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Expectation of Failure Time

One hundred items are simultaneously put on a life test. Suppose the life times of the individual items are independent exponential random variables with mean 200 hours. The test will end when there have been a total of 5 failures. If T is the time…
user63334
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PDF of Ratio of Two Uniformly Distributed Discrete Random Variables

Let $X_1$ and $X_2$ be independent random variable from $\{1,2,3,\dots, n\}$ with PDF $f(x) = 1/n$. What is the PDF of $Y = X_1/X_2$? Here's my attempt The CDF of $Y$ is \begin{align*} F(y) &= P(Y\leq y)\\ &= P(X_1 \leq X_2Y)\\ &= \sum_{x_2 = 1}^n…
Azlif
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How to prove pmf of NBIG >0 and it sum add up to 1?

The pmf of Negative Binomial-Inverse Gaussian is $Pr(X=x)=\displaystyle\binom{r+x-1}{x}\left[\sum_{j=0}^{x}(-1)^{j}\binom{x}{j}\exp\left\{\frac{\psi}{\mu}\left[1-\sqrt{1+\frac{2(r+j)\mu^{2}}{\psi}}\right]\right\}\right]$ how to prove it's pmf always…
Eldominic
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Problem with T distributions.

In the Cambridge HL Maths Statistics & Probability section there is a chapter on t-distributions, however as per usual they do not explain where to begin with most questions. Assuming $T \sim t_7$ ($n=7$), solve $$P(-t < T) + P(0 < T) + P(t < T) +…
John Miller
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The limiting CDF of a Hypergeometric random variable

Let $X_n$ be a Hypergeometric random variable, with parameters $N,K,n$, ($N$ the population size, $K$ the special elements, $n$ the sample size). The PDF is $$P(X_n=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}},$$ and the CDF is $$F_n(x)…
Teddy
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Is there a math law for the random distribution of digits from 0 to 9 of 2^4000 or larger than that

The problem is to calculate the distribution of digits $0-9$ of $2^{4000}$, and the interesting result shows that it is almost a random distribution with almost equally $10\%$ of each digit. After that, I tested $2^{5000}$, $2^{6000}$, and so on,…
L.Maple
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Probability number A greater number B

Given $a \in \{1,2,...,250\}$ and $b \in \{0,1,...,1000\}$ $a$ and $b$ are chosen randomly, how does one calculate the probability of $a > b$?
Meik Vtune
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Converting kelly game to log-normal distribution

Let us assume there is a game in which there is a p probability of success and 1-p of failure and v dollars are bet on each round and the total bankroll is k dollars. The un-invested portion is allowed to compound at some rate $r_0 $. A successful…
Bert45433
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Find the probability mass function of the (discrete) random variable $X = Int(nU) + 1$.

For a non-negative real number $x$, write $Int(x)$ for the largest integer that is less than or equal to $x$. Let $U$ be a uniform random variable on $(0,1)$ and $n \geq 1$ an integer. Find the probability mass function of the (discrete) random…
user59036
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Combining two Beta disributions

I want to combine two beta distributions to find the the posterior distribution as the precision-weighted combination of the prior and the likelihood distributions. As the prior and likelihood I have two beta distributions. How do i combine them?…
Bik
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Finding $\mathrm{Cov}(X,Y)$ and $E(X\mid Y=y)$ given the joint density of $(X,Y)$

The continuous random variables $X$ and $Y$ have joint density function $f_{xy} = 1$ for $0 < x < 1$ and $2x < y < 2$, and zero otherwise. I am stuck on the above question, with parts a and b below: a) Find the $cov(x,y)$. Work: $cov(x,y) = E(XY) -…
peco
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Distribution of distance from origin to ellipsoid

Let $D$ be a $d\times d$ diagonal matrix with positive elements on the diagonal, then $x^\top D x = 1$ defines an ellipsoid in $\mathbb R^d$. Let $p$ be a random point on the ellipsoid in the sense that $p/\|p\|$ is uniformly distributed on…
Mayu
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Prove that $X+Y$ and $|X-Y|$ are uncorrelated random variables but are not independent random variables

Two random variables $Z$ and $W$ are uncorrelated if $E(ZW)= E(Z)E(W)$. Let $X$ and $Y$ be independent random variables receiving 1 with probability $\frac{1}{2}$ and $0$ otherwise. Prove that $X+Y$ and $|X-Y|$ are uncorrelated random variables…
user59036
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Distance between two distributions and measure of points which are closer than the average

I have a question related to distance between two CDF's. Suppose we have CDF $F$ and $G$: An RV $X$ follows $F$ and $Y$ follows $G$, with the same support, say $[0,1]$. If we match each $x$ to $G^{-1}(F(x))$ (matching points that have the same…
Greenteamaniac
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