Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

learn more… | top users | synonyms

5
votes
1answer
505 views

Multivariate Hypergeometric Distribution/Urn Problem

I am having a difficulty with the following multivariate hypergeometric distribution problem. The setting is as usual, an urn contains a total of $M$ balls of $K$ unique colors, with $N_1$ balls of ...
2
votes
3answers
126 views

Which is the probability to a random line to be parallel to a specific other line?

In my perception, using the common sense, is less common, or less probable, to a random line be parallel that not to be, because to be parallel a line needs obey a restrictive rule. But anyone can, ...
1
vote
1answer
72 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
1
vote
1answer
211 views

Well known probability distributions defined on a $n$-dimensional simplex besides the Dirichlet distribution?

Are there well known probability distributions defined on a n-dimensional simplex besides the Dirichlet distribution where the variation of of each component doesn't vary as much when the mean of the ...
0
votes
1answer
54 views

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive ...
1
vote
0answers
464 views

Probability distribution of the sum of products of discrete iid uniform random variables

Here is a problem I am working on, hoping to get some guidance from the experts here: Given the arrays $C=[C_1,C_2,...,C_N]$ and $S=[S_1,S_2,...,S_N]$ of lengths $N$ with elements that are discrete ...
1
vote
1answer
65 views

Variable frequency redistribution

I know that there is a way to "redistribute of the frequencies of a variable" as stated here: Dorian Pyle book chapter 7 section 2 paragraph 3 (7.2.3): The easiest way to adjust distribution ...
1
vote
1answer
352 views

Question about Gamma distribution

For $X$ a gamma random variable with parameters $\alpha >2$ and $\beta> 0$ a) Prove that the mean of $\dfrac1X$ is $\dfrac{\beta}{\alpha-1}$. b) Prove that the variance of $\dfrac1X$ is ...
2
votes
1answer
119 views

Calculating expected number of rounds

I am trying to find the expected number of rounds for a system to finish a process. Lets say N = 100. One process starts off the program by sending a message to another random process. The choice is ...
2
votes
1answer
336 views

How do I calculate the aposteriori probability distribution for someone's answer to a poll being an approval?

Imagine I'm polling a random sample from the population and it asks them if they approve of the President or not. I also ask them some categorical demographic questions (age-bracket, race, gender, ...
2
votes
0answers
100 views

How to sample from a product-of-sums distribution?

$A$ is a $M$x$N$ matrix whose entries are positive. $x$ is a $N$ dimensional binary (i.e. consisting of $0$s and $1$s) vector and the number of $1$s in $x$ is constant. Let $y = Ax$. The distribution ...
2
votes
1answer
225 views

Conditioning on zero probability event

I have a somewhat trivial question (no homework): Suppose $X_1, X_2$ are i.i.d. and uniform on $[0,1]$, and the realization of their maximum is $k \in (0,1]$. What is the conditional distribution of ...
1
vote
0answers
50 views

Stationarity of an Integral process

Let $f$ be a continous deterministic function defined on $[0,c]$ and $(B^{H}_{t})_{t\geq 0}$ be a fBM with $H\in(0,1)$. We define a Process $ (X_{t})_{t\geq 0}$ with ...
0
votes
1answer
404 views

Whats the probability that a certain amount of batteries life longer than 3.25 hours?

The mean and standard deviation of the lifetime of a battery in a portable computer are 3.5 and 1.0 hours respectively. So, what is the probability that the mean lifetime of 25 batteries exceeds 3.25 ...
4
votes
2answers
2k views

Conditional probabilities involving the exponential distribution

The number of years the laptop functions is exponentially distributed with mean = 5 years. If a customer purchased an old laptop which was used for last two years, what is the probability that it will ...
4
votes
2answers
528 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
6
votes
1answer
343 views

How to prove this combinatorial identity?

I am wondering how to prove the following identity: $$\sum_{i=0}^{n-r} \frac{2^i (r+i) \binom{n-r}{i}}{(i+1) \binom{2n-r}{i+1}}=1?$$ It seems this might be related to the hypergeometric distribution, ...
0
votes
1answer
121 views

Reversibility of Markov Process and Exponential Distribution of Transition Rates

I am reading the paper Towards Utility-optimal Random Access Without Message Passing by J. Liu, Y. Yi, A. Proutiere, M. Chiang, H. V. Poor. A sentence in Section 3.2 can be paraphrased as follows: ...
1
vote
0answers
165 views

Cross-section of a circle with a three-dimensional Gaussian

Suppose I have a three-dimensional Gaussian with mean $\bar{\mu}$, volume $A$ and covariance matrix $\Sigma$ $$G(X)=\frac{A}{\sqrt{(2\pi)^{3}\det(\Sigma)}}e^{-\frac{1}{2}(X-\mu)^{T}\cdot ...
0
votes
1answer
477 views

Probability for Communication Networks

A computer communication channel transmits words of $n$ bits using an error-correcting code which is capable of correcting errors in up to $k$ bits. Here each bit is either a $0$ or a $1$. Assume ...
0
votes
1answer
71 views

Probability distribution for a function of a random variable

I have the distribution of X with respect to parameter t vaying between 0 and 1. However, in nature, parameter t is not uniformly distributed. It has a known probability distribution. What is ...
3
votes
1answer
188 views

A limitation related to multinomial distribution.

recently I have a problem about the multinomial distribution. Here, for positive integer $n$, $$ t_{n}=\sum_{i=1}^{n}a^{i}\sum_{i_{1},\ldots i_{n}}\left(\begin{array}{c} i\\ i_{1},\ldots i_{n} ...
0
votes
1answer
276 views

Help solving CDF for transformation of $ \ge 2 $ random variables or if it's impossible.

Suppose independent random variables $U, T$. Let $U$ have continuous Uniform distribution over $(0, 2\pi)$. Let $T$ have Exponential distribution with $\lambda = 1$. Let random variable $Y$ be a ...
6
votes
3answers
6k views

How exactly are the beta and gamma distributions related?

According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation: $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$ Can you point me to a derivation of this ...
0
votes
1answer
64 views

Optimization with respect to distribution mean involving error function

I have been working on the following problem for some time now, to no avail, so any advice whatsoever is greatly appreciated. I would like to optimize $\min_{0 \leq r \leq 1} \ \ L = ...
3
votes
5answers
4k views

What is the use of moments in statistics

Can any one give an "simple" explaination about what is the use of moments in statistics.Why we need moments? what we can learn from it? if possible please use less equations. Advance thanks for your ...
0
votes
2answers
1k views

Adjusting mean and standard deviation

There's a set of 8 bags with the following weights in grams given: 1013, 997, 1013, 1013, 1004, 985, 991, 997 The mean is 1001.625, unbiased standard deviation is 10.86. I have the following ...
1
vote
3answers
212 views

What is wrong with this approach to find Expected value of distance between $X, Y \in$ Uniform $(0, 1)$?

Let $X$ and $Y$ be i.i.d. random variables with Uniform $(0, 1)$ continuous distribution. The problem is to find the expected value of the distance between X and Y. My reasoning was, for all $(x, y) ...
1
vote
0answers
601 views

What is the significance of error function?

Here's a Wiki article on the subject. Sadly it doesn't do a good job of explaining the significance of the function. Of course it may mean different things to different people (for mathematicians it ...
1
vote
3answers
902 views

What is the maximum expected value in a selection?

Here's a hypothetical problem: assume that mean diameter of a tennis ball is 6.7 cm. Assume that the diameter is normally distributed with a standard deviation of 0.1 cm (I may have picked up a weird ...
1
vote
1answer
184 views

Entropy of Zipf and Zeta Distributions

I was wondering how to show entropy of the zeta distribution. It is: $$ H_\mathrm{zeta}(X) = \sum_{k=1}^\infty \frac{1/k^s}{\zeta(s)} \log(k^s \zeta(s))$$ The entropy of the zipf distribution is: ...
2
votes
2answers
108 views

Computing covariance with the help of the memoryless property

Question: Suppose that in a certain town earthquakes occur as a Poisson point process with an average of 3 per decade, floods are a Poisson process with an average of 2 per decade, and meteor strikes ...
3
votes
1answer
631 views

Conditional Moment Generating Function With A Twist

Let $X$, $X'$ be identically distributed (not necessarily iid) random variables with compact support, on the same probability space. Define $G_t(x):=\mathbb{E}[e^{t(X'-X)} | X=x]$ In other words a ...
1
vote
0answers
94 views

continued fraction multivariate normal distribution?

After searching for a while, I wonder if a continued fraction representation exists for the multivariate Mills ratio $P(Z \geq x)/\phi_Z(x)$ where $Z \tilde\, N(\mu,\Sigma)$. The representation ...
2
votes
1answer
115 views

How do you take the product of Bernoulli distribution?

I have a prior distribution, $$p(\boldsymbol\theta|\pi)=\prod\limits_{i=1}^K p(\theta_i|\pi).$$ $\theta_i$ can equal $0$ or $1$, so I am using a Bernoulli distribtion so that ...
2
votes
1answer
78 views

Find the probability of one expression

Could you please help me to calculate the probability as follows \begin{align} \rho=\mathsf{Pr}\left\{\frac{X_1}{X_2} >\frac{Y_1+a}{Y_2 + a}\right\} \end{align} where $X_1,X_2, Y_2$, and $Y_2$ ...
0
votes
3answers
639 views

Finding \alpha and \beta of Beta-binomial model via method of moments

I am looking for a laymen step by step of how the process of finding the 1st and 2nd sample moments located: http://en.wikipedia.org/wiki/Beta-binomial_distribution#Maximum_likelihood_estimation ...
11
votes
3answers
625 views

Random point uniform on a sphere

If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$, Are the coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$?
1
vote
2answers
1k views

minimize variance

$X_1$ and $X_2$ are independently distributed random variables with $$P(X_1=\Theta+1) = P(X_1=\Theta-1) = 1/2 \\ P(X_2=\Theta-2) = P(X_2=\Theta+2) = 1/2$$ Find the values of a and b which minimize ...
1
vote
2answers
3k views

Finding distribution function of $Y/X$ and probability density function of $X+Y$

I'm studying for an exam at the moment, and these types of questions have just got me stumped to the point where I need a step-by-step walkthrough... More specifically I've got two questions I just ...
0
votes
0answers
69 views

Plot randomly oriented gaussian kernel

I would like to plot with scipy randomly oriented gaussian kernels. For a gaussian kernel along x and y axis (with an angle 0 w.r.t. coordinate system), I simply plot function ...
4
votes
1answer
364 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
2
votes
2answers
213 views

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
5
votes
2answers
6k views

What is the correct inter-arrival time distribution in a Poisson process?

Given a Poisson process (e.g. radioactive decay) with rate $\lambda$, then the expression $\exp(-\lambda t)$ is the probability of observing no counts in time interval $t$. This can be interpreted ...
1
vote
0answers
228 views

Polynomial approx to the Normal density

I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF? Thanks $^{(1)}$ For example, some are ...
1
vote
1answer
103 views

Probability Density Function: Small question

Let $X \sim N(\mu, \sigma^2)$ and $f_{X} {(x)} = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)$, where $-\infty < x < \infty.$ Find the PDF of $X^3$. Do you ...
2
votes
2answers
372 views

Probability distributions - Exam paper question - $\mathrm{Cov}(X,Y)$, PDF

Two people have decided to meet at a certain point in a forest sometime between noon and 2pm. Their respective independent arrival times are $X$ and $Y$ such that $X \sim \mathrm{Unif}(0,2)$ and $Y ...
1
vote
1answer
1k views

Characteristic function of a sum of Uniform random variables

Suppose I have $S=U_1+U_2+\dots+U_n$ where $U_i$ are distributed Uniform$[-1,1]$. I am trying to show a couple of things. First, what is the characteristic function. I can show this easily enough for ...
1
vote
2answers
65 views

What are some heuristics to minimize how many states to choose to maximize the cumulative probability in a discrete probability distribution?

Obviously, we could just choose all the states and get 100% probability. That would maximize the probability, but not minimize the number of states we'd have to choose. The other extreme would be to ...
2
votes
1answer
394 views

Find the probability that the second customer to arrive has to wait to be served if arrival time is exponential and serving time is uniform

Customers line up to be serviced according to a Poisson process at an average rate of five per hour. If the time it takes to serve one customer is a continuous uniform random variable on $[0,4]$, ...