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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Maximum possible distance between two vectors sampled from n-variate Gaussian

What would be the probability distribution of the distance between two vectors sampled from n-variate Gaussian distribution? Thanks.
user138617
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A probability question regarding two independent uniform distrbutions.

I am thinking about this question: $X_1$ and $X_2$ are independent $Unif(0,1)$ random variables. (a) Derive the pdf of $\overline{X}=\frac{X_1+X_2}{2}$. (b) Calculate $E(\frac{X_1}{\overline{X}})$. (c) Calculate $E(X_1|\overline{X})$,…
fblues
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What's the distribution of the difference of two gamma distributions? (the two gamma have same shape parameters)

If $X∼\textrm{Gamma}(a,b_1)$ and $Y∼\textrm{Gamma}(a,b_2)$, $X$ and $Y$ are independent, what is the distribution of $X-Y$?
user258682
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Sufficient condition for monotone likelihood ratio property

What are sufficient conditions for the monotone likelihood ratio property? I have a set-up where $F(x)$ (cumulative distribution function of r.v. $x$) always exceeds $G(x)$ (a different cum. distrib. function), when these c.d.f.s are taken as…
William Grove
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What is the distribution of $\frac{X_1}{X_2}$ if both $X_1$ and $X_2$ follows the Poisson Process?

What is the distribution of $\frac{X_1}{X_2}$ if both $X_1$ and $X_2$ are the Poisson Processes with parameters $\lambda_{1}$ and $\lambda_{2}$ respectively? Please list the properties used too. Thanks !
user2778430
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Poisson process - number of store purchases in a given time

Customers enter a store according to a Poisson Process of rate = 6 per hour. Individuals who enter the shop have (independently of each other) probability $\theta$ of buying something. If exactly n people enter the shop during a certain…
user123965
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Poisson distribution question about traffic lights

Question: on the way to work a guy must pass through $10$ traffic lights suppose that in the long run he encounters a red light at $40$% of these signals and whether any particular signal is red is independent of whether any other one is red On what…
Arvin
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Limits of the joint pdf $(2/3)(x + 2y)$

I'm given equation that the joint pdf is $(2/3)(x + 2y)$ when $0 < x, y < 1$ and we want to find the probability that $X < 1/3 + Y$. I understand how to do the actual math part, and that I have to do a double integral. But what are the inner and…
Shell
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${\bf E}[Y]$ of a joint distribution

So, I have that a joint probability density function is given by the formula: $$ 5e^{-5x} / x, \quad 0 < y < x < \infty $$ and I have to find the $\operatorname{Cov}(X,Y)$. I know that $\operatorname{Cov}(X,Y) = {\bf E}[XY] - {\bf E}[X]{\bf…
Shell
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Joint and marginal distributions of independent uniformly distributed variables

Suppose that $X_1$ and $X_2$ are independently uniformly distributed on the interval (0,1). Find the joint and marginal distributions of $U=X_1X_2$ and $V=X_1/X_2$. I think that $f_U(u) = \int^1_0f_{X_1}(x_1)f_{X_2}(u/x_1){1\over{|x_1|}} dx_1$ I'm…
user123965
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Probability Distribution of Runs in Coin Flips

If you flip a coin $n$ times, what is the probability distribution of the longest "run" (sequence of consecutive heads or tails) which will occur? Or if that's not possible, what is the average? I have tried everything I know of and have never been…
user142299
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Joint p.d.f and independence

I kinda remember there is a result like this from Probability theory, but I forgot how to prove it. Is there a formal name for it? Can someone kindly provide me with the proof or a link please The random variables $X$ and $Y$ with density $f$ are…
user1819047
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Proper name for this distribution type

I need help in defining the type of distribution used in software so that I could use some standard distribution library for this purpose. I apologize for not using proper terms. It takes "center" value and an alpha parameter (0 to 1) and produces N…
Boris Mikhaylov
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Minimum of identical independent Poisson random variables

Assume $X_1,X_2,\ldots,X_n$ are identical independent random variables, all distributed with $\text{Poisson}(\lambda)$ (same $\lambda$). I am interested to find out the distribution of their minimum: $X_\min=\min_i\{X_i\}$. Especially I am…
Bach
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Limit of binomial distributions whose expectation tend to 0

According to the Poisson limit theorem, if $np\to\lambda$, then $\text{Bin}(n,p)\to\text{Poisson}(\lambda)$ (all when $n\to\infty$). Does that mean, in particular, that if $np\to 0$, the limit distribution is the atomic distribution of 0? That is,…
Bach
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