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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Question on finding the median of a discrete random variable

I have been trying to solve this math problem, but I end up with a different answer, I suspect the publisher probably used a different approach but I am not sure which one. Here is my attempt: My cumulative distribution function, $F(x) = P(X \leq…
Meghan C
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Finding Density Function of a Function of other Random Variables

Suppose that two electronic components in the guidance system for a missile operate indepen- dently and that each has a length of life governed by the exponential distribution with mean 1 (with measurements in hundreds of hours). Find the…
Bryden C
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Marginal density of X error in the test?

So I took a test. And it was easy, but for some reason I got 0pts out of 5pts in a once single exercise. OK, here it is: "Given the joint density function $$f(x,y)=12/7$$ when $$0\leq x \leq 1 \text{ and } 0\leq y \leq 1$$ And 0 elsewhere.…
NabbKitha
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Convolution of two independent exponential random variables

Let $X\sim \text{Exponential}(\frac{1}{\lambda})$ and let $Y \sim \text{Exponential}(\frac{1}{\mu})$. Let $Z = X+Y$. I want to find the pdf of $Z$. I start by using the convolution $$f_Z(z) = \int_{-\infty}^{\infty} f_X(z-t)f_Y(t)dt$$ $$f_Z(z) =…
mrnovice
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Mapping the uniform distribution to an inverse power distribution

I have looked up a lot about how to do inverse transform sampling and still cannot figure this out. I have an inverse power distribution $f(x)=\frac{1}{(x+1)^n}$ where $n$ is an integer greater than 1. I've done the integral from 0 to infinity and…
Radagast
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Special case of distribution function

I want an example for a distribution function $F$ with those two conditions for all $\lambda>0$ $$\liminf_{x\to\infty}\frac {1-F(x)} {e^{-\lambda x}}=0 $$ $$\limsup_{x\to\infty}\frac {1-F(x)} {e^{-\lambda x}}=\infty$$ I want one example of a…
nordmann
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Skewness and kurtosis of sum of lognormals

Below is an equation $f(x)$ that is the sum of two lognormal waveforms. All of the coefficients are known, and by integration, the area under the curve is simply $P_1 + P_2$. $\displaystyle f(x) = \frac {P_1 } {S_1(x + D) \sqrt { 2 \Pi}} e ^…
zax
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moment generating function of a summation

Suppose the time between independent incoming calls to a call center is well modeled by an exponential distribution having $\lambda = 0.025$. Let $T_i$ be the time between the $i$th and the $(i-1)$st call, and define $S_k = \sum^k_{i=1} T_i$. What…
hello888
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Binomial distribution: Suppose that a basketball player sinks a basket from a certain position on the court with probability $0.35$

2.3.15 Suppose that a basketball player sinks a basket from a certain position on the court with probability $0.35$. (a) What is the probability that the player sinks three baskets in 10 independent throws? (b) What is the probability that the…
Bas
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Distribution of random variable 'Time to n-th maximum'

Let $\{X_1, X_2, ...\}\overset{iid}{\sim}~\mathcal{D}$ is a time series of iid random variables drawn from a continuous distribution $\mathcal{D}$. How can I calculate the distribution of a random variable $T(n)$, the time to the $n$th maximum,…
Ape
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Compound distribution of Uniform and Exponential

I am trying to take the compound distribution of a uniform distribution on the interval $(0, \frac{t}{2})$ where $t$ is parameterized by the exponential distribution $\varepsilon (\lambda)$. Taking the compound distribution I believe gives the…
Philip Lassen
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Finding the PDF of a product of two r.v.s

Let $X,Y \sim \operatorname{Unif}(0,1)$ be independent of each other. Find the PDF of $V:= XY$. This is was I did $$\mathbb{P}(V \leq v) = \mathbb{P}(YX \leq v) = \mathbb{P}\left(Y\leq \frac v X \right) \\ = \int_0^{v/x} 1 \, dy = \frac v x.$$ I'm…
Mr. Bromwich I
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Prove that memorylessness of a discrete distribution defines geometric distribution

The task is to proof that a discrete distribution is memoryless if and only if it is a geometric distribution. I haven't wrote this proof myself, but I understand how to prove that a geometric distribution is…
gaazkam
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Cumulative distribution function, Variance.

Just need some answers checking: (iii) im not sure what needs doing but i've shown my attempt A continuous random variable $X$ has cumulative distribution function $$F(x) = \begin{cases} 0 & x<1,\\ x^2(3-2x) & 0\le x \le 1,\\ 1 &…
Matt
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Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$?

Let $X$ and $Y$ be independent, standard normally distributed random variables ($\sim Normal(0, 1)$). Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$? I understand that an exponential random variable describes the time until a next…
John Hoffman
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