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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.

28080 questions
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Is there an example of an exponential family distribution with intractable normalizing constant? If so, what is it?

Typically, one comes across examples of exponential family distributions that have analytically computable normalizing constants. Consider the normal, beta, Poisson, etc. distributions. However, I don't see why the normalizing constant would need…
ashman
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What does moment of functions tells you?

What does the nth moment of a function tell us about that function?. Like in the case of distributions, 1st moment tells about the expected value as we are taking the weighted average. But what does the 2nd moment of the function tells us? On its…
Amarjeet Kapoor
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Probability bookmakers game

I have a simple game which works as follows: There are 11 options where you can pick from, they all have a starting score of 100. When a random option is chosen this score is lowered by X, the other options are raised by Y. This makes sure that…
Roger Far
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Give example of a distribution.

Give examples of distribution (1) such that $X$ and $1-X$ have the same distribution. (2) such that $X$ and $\dfrac1X$ have the same distribution. For the first one I think $X$ is $\text{Uniform}(0,1)$. Since $1-X$ is also $\text{Uniform}(0,1)$. I…
A.D
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Phase-type distribution - moments

I was doing my best to calculate moments of phase-type distribution. Density of phase-type distribution is $$f(x)=\alpha e^{Sx}S_{0}$$ ($\alpha$ is $1\times m$ vector; $S_{0}$ is $m\times 1$ vector; $S$ is $m\times m$ matrix) for all $x>0$, where…
Paweł Orliński
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How to Determine the Likelihood of a Specific Input to a Probabilistic Function

Here's the problem: A probabilistic function $f$ takes as its argument an input $x \in X$ with vector output $\mathbf {y} = f(x)$, where other inputs from $X$ might also produce $\mathbf y$, but likely with different probabilities. Now, using only…
Optimalist
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Why are r.v $X$, $Y$ independent given that their joint pdf factors as follows: $f(x, y) = k(x) g(y)$?

I am currently trying to wrap my head around why given the r.v. $X$, $Y$ and their joint pdf $f(x, y) = k(x) g(y)$ we can say that the r.v. are independent. I can see that the marginal distribution function of X is $f_X(x) = k(x)…
Tudorosu
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How can I find the distribution of the discount factor?

What is wrong here?? Suppose that the random variables $Z_i$ are defined as follows: \begin{equation} Z_i = D(0, t_i)(R_{i-1} +c)\Delta N, \end{equation} where $D(0, t_i)= \exp\{-\int_{0}^{t_i} r_u du\}$ for which $r_u$ follows a CIR model,…
Statistics
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Find the distribution of $Y=\sqrt X$ when $X$~$N(0,1)$

PDF of $X$ is $f_x(t)=\frac{1}{\sqrt{2pi}}e^{\frac{-x^2}{2}}$. I tried to found the CDF of $Y$: $F_Y(t)=P(Y \leq t)=P(\sqrt X \leq t)=P(X \leq t^2)=F_X(t^2)$ How do i get distrbution of Y from this? If I were to find the $P(X\leq t^2)$ I would get…
HiThere
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Borel-Tanner distribution with finite bound

I'd like to model proportion of certain species in a popualtion with Borel-Tanner distribution: $\frac{e^{-m}m^{m-1}}{m!}$, its support is defined on $\{1,2,...\}$, but I need finite bound. Could anyone help me with finding the finite sum…
sigma.z.1980
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Generalised solution for Amoeba's survival problem

A population starts with a single amoeba. For this one and for the generations thereafter, there is a probability of $p$ that an individual amoeba will split to create two amoebas, and a $(1-p)$ probability that it will die out without producing…
Elin Das
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Approximating a Poisson distribution to a Normal distribution

I have the following problem I'm trying to solve: I know that the quantity of complains in a call center is a Poisson variable with $\lambda=18 $ costumers/hour, and that the probability of being able to solve a complain is $0.35$. They ask me about…
FDrico
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Poisson Distribution - sum of RVs

Question: $X$ balls are thrown to $n$ bins (each ball has an equal chance to get to each bin). Let $X_1,\dots, X_n$ be the amount of balls in each cell. a. Show that if $X \sim \text{Poisson}(\lambda)$ then $\displaystyle X_i \sim \text{Poisson}…
jreing
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If $X$ is a random variable and $a$ is a positive number then does $Y = \max (a,X)$ is a legitimate random variable?

If $X$ is random variable with a given PDF and $a$ is a positive number then could $Y = \max (a,X)$ be considered as a legitimate random variable ? Mathematica result look strange ! For example, when $Y = \max (3,X)$ and $X$ follow the exponential…
Tuong Nguyen Minh
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Weibull parameters as a function of values of the c.d.f.

Let $X$: The waiting time between two earthquakes (in years). We know that $X$ is following a Weibull distribution, i.e. $X \sim W(\gamma=0, \beta, \delta)$. How can I find $\beta$ and $\sigma$ knowing that $P(X>15) = 0.3679$ and $P(X\leq 5) =…
J.Doe
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