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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.

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Find $g(x)$ such that $f(x)/g(x)$ is non-decreasing

Does anyone know what the possible probability density function (pdf) are that $g(x)$ can take such that $f(x)/g(x)$ is non-decreasing, where $f(x)$ is a mixture of 2 normal densities? For example, take $$f(x)=\frac{1}{2}N(0,1)+\frac{1}{2}N(3,1).$$…
user340803
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Approximating a sum of two binomial distributions

A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams, and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with…
woaini
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Minimum of random variables with a discounting term

I have the following problem. Assume that $X_t$ are all i.i.d. random variable uniformly distributed between 0.5 and 1.5, and that $\phi$ is a little larger than 1. $$Y_t= \min(X_t, Y_{t-1} \phi)$$ Which of course can be written as $$Y_t= …
Andrea
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Calculate the probability of the sum of multiple discrete independent variables

There are $N$ independent discrete variables $\{x_i\}$, $i=1,2,\cdots,N$, where $\Pr(x_i=1)=p_i$ and $\Pr(x_i=0)=1-p_i$. Let $K=\sum_{i=1}^{N}x_i$. Then, how about the probability $\Pr(K=k)$?
Dave
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Joint Probability Distribution Involving Arrival Times

Bob is to meet Joe between 11:30 am and noon. If they arrive at random times during this interval and their arrivals are independent, what's the probability that they'll have to wait for each other at most 10 minutes? I understand that the events of…
blargen
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Geometric Laplace distribution

I'm working through Laplace distributions within a stock model. The stock price is modelled with a geometric Laplace motion along the lines of $(1-\frac{\sigma^2}{2}\cdot dt)*exp(r*dt-\sigma*\sqrt{dt}*f_L)$ def laplacestock(r, sigma, mu, dt = 1): …
Huang_d
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Get Joint PDF from Joint CDF

Let joint cumulative probability density function of Random variable X,Y $$F_{1,2}(x,y) = x^2(1-e^{-2y})\;\; \text{when}\;\;0\le x\lt1, y\ge0$$ and $$= (1-e^{-2y}) \;\; \text{when}\;\; x\ge 1, y\ge0$$and $$=0 \;\; \text{when} \;\;y \lt 0$$ in this…
Beverlie
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What is the pdf and cdf of $Y := XD +(1-X)P$ where $X \sim Ber(p)$, $D\sim Exp(\lambda_d)$,$P\sim Exp(\lambda_p)$?

What is the pdf and cdf of $Y := XD +(1-X)P$ where $X \sim Ber(p)$, $D\sim Exp(\lambda_d)$,$P\sim Exp(\lambda_p)$? All variables are independet. Thanks for the hint. By the Hint…
user276611
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Why we need Probability-generating function or MGF?

This is a basic question, but I barely find solution about it. I found there are a lot of explanations or examples about def, calculation or properties. Any master can give me some example about why we need ? Again, the question is WHY we need PGF…
sunson29
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Definition spherical distribution

I read in serveral books about the definition of a spherical distribution; $x \in \mathbb{R^n}$ has a spherical distribution if and only if $x \stackrel{d}{=}Ox$ for any orthogonal, $n$-dimensional matrix $O$. But now I try to understand that;…
user299124
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Two Continuous Dice

I assume two continuous independent random variables $x$, $y$ with the same PDF $p(x)=\frac{A}{|x|^a} (|x|c)$ and $\frac{1}{2}
richard
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Two points are chosen uniformly on a unit circle. What is the distribution of the shortest arc length between them?

Two points, $A$ and $B$, are independently and uniformly selected from a unit circle. What is the distribution of the shortest arc length between them? The answer is supposed to be that the shortest arc length is distributed as a uniform…
user418749
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derived distribution of random variables

How can I derive (even approximately) the distribution of the random variable $$\Delta=\cos\phi_1-\cos\phi_2$$ if I know the following two facts about the random variables $\phi_1$ and $\phi_2$? 1) $\phi_1-\phi_2\approx \mathcal{N}(0,\tau)$ (i.e.…
Ziofil
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What is the expression for kurtosis(X - Y)

The skewness of the difference between two independent random variables, X and Y, is given by: $$\text{skewness}(X-Y) = \frac{\mu _3(X) - \mu _3(Y)}{\big(\mu _2(X) + \mu _2(Y)\big)^{3/2}}$$ But what is the expression for kurtosis(X - Y)?
user120911
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Binomial probability for large $n$, small $p$

I need to compute the probability of getting more than $x$ "successes" in a large number of trials $\left(\,10^{11}\,\right)$ of an event with a small probability $\left(\,10^{-7}\,\right)$. Exact Binomial won't work, and the Poisson approximation…
Dmitri
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