Questions about performing probabilistic calculations, especially those concerned with the Mathematica commands Probability, Expectation and the various functions related to distributions such as NormalDistribution, CDF, PDF etc.

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30
votes
6answers
2k views

Efficient way to generate random points with a predefined lower bound on their pairwise Euclidean distance

Using Mathematica what is an efficient way to generate a list of $n$ random two dimensional points $\{x_i,y_i\}$ where $i=1,...,n$ so that no two points $p_1$ and $p_2$ in the list has an Euclidean ...
28
votes
1answer
2k views

RandomVariate from 2-dimensional probability distribution

A probability distribution can be created in Mathematica (I am using 8.0.1) with e.g. ...
20
votes
3answers
355 views

Negative probability?

Bug introduced in 9.0.1 and fixed in 10.0.2 I am trying to get the sum of the squares of seven random variables, all uniformly distributed. This is what I tried. ...
17
votes
3answers
1k views

What is the average of rolling two dice and only taking the value of the higher dice roll?

I am referencing to this question on mathunderflow: What is the average of rolling two dice and only taking the value of the higher dice roll? Now I tried: ...
16
votes
2answers
857 views

How to compute the inverse CDF properly?

Note: this has been fixed in version 9. I want to compute the CDF and inverse CDF of the hyperbolic distribution: ...
14
votes
2answers
1k views

Obtaining joint distributions and conditional distributions using Mathematica

I have two multivariate Gaussian distributions $p(x)$ and $p(z)$ with mean vectors $m_x$ and $m_z$, and covariance matrices $\Sigma_x$ and $\Sigma_z$. my model is a simple linear model $x = W z+n$ ...
14
votes
1answer
348 views

RandomVariate returns values outside the support of a PDF

Let $X$ be a random variable with pdf: dist = ProbabilityDistribution[1/(Abs[x]*Log[Abs[x]]^2), {x, -E^-2, E^-2}] Here are some pseudo-random drawings from it: ...
13
votes
1answer
264 views

Randomness test suites that can be invoked from Mathematica

There are several test suites referenced at csrc.nist.gov designed to test a (pseudo-)random byte sequence for randomness, and discover possible skews, correlations, patterns, or other hidden ...
12
votes
3answers
487 views

Simulate a simple spinner

I'm teaching some simple ideas in probability to students in grade 7. The "spinner" below works, but I'm wondering how I could make it just a bit more realistic by having it actually "spin" around a ...
12
votes
1answer
173 views

NExpectation behaves oddly with EmpiricalDistribution

Note: this is fixed in version 9. My question concerns the usage of NExpectation and Expectation and why I see the behavior I ...
11
votes
6answers
1k views

2D Gaussian distribution of squares coordinates

I would like to imitate the structure of this great painting from Ellsworth Kelly in Mathematica. Yet with all the colored squares in Black and the beige one in white. Below is what I have wrote to ...
11
votes
2answers
3k views

How to calculate the mode of a probability distribution

I was wondering if there is any command to get the mode of the probability distributions. Calculating manually, the mode is (alpha-1)/(alpha+beta-2) While the mean, median or variance of the beta ...
11
votes
3answers
3k views

How to generate a RandomVariate of a custom distribution?

I'm trying to generate a pseudorandom variate out of a custom distribution. Suppose I want define a custom distribution, and for the sake of simplicity I define a Poisson distribution (the ...
10
votes
4answers
547 views

How do I solve this probability problem with Probability´╝č

Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What is the probability that the three segments obtained in this way form a ...
10
votes
4answers
424 views

Extracting equations from Piecewise expressions

Say I have a PDF: PDF[LogNormalDistribution[1.75, 0.65], x] Calculating it, Mathematica gives me an expression that looks like this: I want to extract the ...
9
votes
2answers
975 views

Mathematica Package for Bayesian Networks

Are there any packages that allow the simulation of Bayesian Networks with Mathematica? I found what seemed to be a promising package (Dynamics) on a Brown University URL, ...
9
votes
1answer
872 views

Mathematica code for hidden Markov models (HMM)

I am looking for some simple Mathematica code to model an HMM with just a few states and an equal number of observable signals (emissions). I am hoping to generate sample paths and keep track of the ...
8
votes
3answers
533 views

Probability problem — Rube Goldberg solution?

A user posted this question on StackOverflow which was closed as off topic: 3 people are playing a game with a standard 52 card deck. Each player is given 2 cards each, possible cards and their ...
8
votes
2answers
570 views

Expected Value: Strange/unexpected result

Assume $X$ is an exponential random variable with unit mean ($f_X(x)=e^{-x}$, $x>0$). I want to calculate the expected value of $\frac{1}{X}$. Since $X>0$, I am expecting that ...
8
votes
1answer
418 views

Random variables with transformed discrete distributions cannot be applied to Probability[] function?

I have two discrete uniform distributions over same support: Dx = DiscreteUniformDistribution[{-8, 8}]; Dy = DiscreteUniformDistribution[{-8, 8}]; I use ...
8
votes
1answer
585 views

Recommended book on random processes to understand new functionality in Mathematica 9?

I am interested in exploring the new functionality on random processes available in Mathematica 9, but I am not familiar with all of the underlying mathematics. Could you recommend a book that ...
7
votes
1answer
1k views

Probability and distribution from actual data

Let's say I have some data from a real world system: ...
7
votes
2answers
176 views

Why is mathematica giving wrong and incomparable results for the integral?

1) Integration of Gaussian Distribution with $(x,y,z)$ ranging from $-\infty$ to $\infty$ gives 1 as expected using this command in mathematica. (Total Probability = 1) $\sigma = 200000$ and ...
7
votes
1answer
163 views

RandomVariate does not fill range of PDF of distribution

I've declared my own probability distribution as follows dist[a_, b_] := ProbabilityDistribution[Cos[(b x)/2]^2 Sinc[a x]^2, {x, -400, 400}]; When I try to ...
7
votes
1answer
167 views

How to create a spliced survival statistical distribution?

I am trying to use Mathematica for more of my reliability engineering work and am having trouble using SplicedDistribution. I have a Weibull distribution (dist1) associated with a low stress ...
7
votes
1answer
621 views

How do I calculate the probability of reaching mean residual life

I need to understand the Survival Analysis concept of "mean residual life (MRL)" and calculating probabilities for reaching it. From another discussion: The MRL at time t is the mean additional ...
7
votes
0answers
165 views

Calculating probabilities symbolically

Is there a way to solve for statistical quantities analytically/symbolically in Mathematica? example 1: Lets say that I want to do a calculation that requires Bayes theorem. I know p(a), p(b) and ...
6
votes
2answers
2k views

Function to compute the probability of exactly one event occurring out of N independent events

Is there a built in function (or a function in one of the standard packages) that allows you to compute the probability of exactly one event occurring out of some known set of probabilities for N ...
6
votes
3answers
476 views

Calculate probability function

I would like to calculate the following with Mathematica: Let $X$ be a random variable with probability function $$p(x)= \left\{ \begin{matrix} \frac{x}{6} & \text{for }x \in \{1,2,3\}\\ 0 ...
6
votes
1answer
132 views

How can I create a Distribution from its probability density function?

I have a probability density function. Is there a mechanism to create a Distribution object from it, such that normal Mathematica commands that usually work on ...
6
votes
4answers
260 views

Conditional Expectation — How can Mathematica find a more general closed form?

Mathematica can't find a solution to this Expectation. ...
6
votes
1answer
99 views

Why am I getting wildly incorrect results from FirstPassageTimeDistribution with inexact transition matrix?

Working on some large systems using DiscreteMarkovProcess, I changed the transition matrix to machine precision vs using exact values, which sped things up handily. ...
6
votes
1answer
248 views

Speeding up the non-negativity algorithm for multilinear function with interval probabilities?

The algorithm is a branch-and-bound algorithm that calculates the non-negativity of a multilinear function with interval probabilities. The lines 9-11 has an optimization that is explained on the ...
5
votes
4answers
323 views

Probability: proportion of 1000 random lists for x that contain the same nrs

Give a function which calculates for each x a random list of birthdays (which can be x=1 to x=365) birthdays[x_] := RandomInteger[{1, 365}, x] is my answer. ...
5
votes
1answer
520 views

Is there a free version of the Springer book “Mathematical Statistics with Mathematica”?

Is there a free version of the Springer book: $$\text{Mathematical Statistics with Mathematica}$$ by Rose and Smith ... available on the internet?
5
votes
2answers
135 views

Getting the Cumulative Distribution Function of f(x) from a dataset of x

I've a data set $\mathcal{X}$ which consists of randomly generated numbers. Let's say for example data = RandomVariate[ExponentialDistribution[1], 10^4]; My aim ...
5
votes
1answer
277 views

How to solve this probability symbolically or numerically?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim ...
5
votes
1answer
327 views
5
votes
1answer
70 views

Speedup of KL Divergence by Monte Carlo Simulation

I started with Mathematica about just a week, so I don't know how to take all of this environment. What I want to know is if there is a way to speedup my code to compute the KL-Divergence of Gaussian ...
5
votes
1answer
167 views

Simulation Sample Paths for Stochastic Approximation Algorithm

Let $X_t$ be IID random variables in $[0,1]$ and CDF $F(\cdot)$. Suppose there exists a variable $Y_t$ given by: $Y_1 = X_1$ and $Y_t = \phi(X_t, \overline{Y}_{t-1}), \forall t >1$, where ...
5
votes
1answer
115 views

MarginalDistribution with Symbolic range in ProbabilityDistribution

Given the following joint density: for 10 < x < 20, x/2 < y < x. To find the marginal density for X we do: ...
4
votes
2answers
239 views

How to find the variance under the assumption that x follows some probability distribution

I am aware that we can find the expectation under the assumption that x follows some probability distribution, something like this: ...
4
votes
4answers
534 views

Generate random points in a region with higher probability in a sub-region

I am trying to generate random samples of points in the unit square in such a way that points inside a given circle are twice as likely to be selected as points outside of that circle. I tried to ...
4
votes
3answers
652 views

Graphical Plots of PDF and CDF

I have a joint density and distribution function that I want to plot in a meaningful way, (i.e., want to be able to see how the functions behaves as changes in x and y happen simultaneously. If ...
4
votes
1answer
207 views

Reproducing a graphic on Jensen's inequality for the probabilistic case

I am trying to reproduce the following graph on Jensen's inequality for the probabilistic case (source: wikipedia) but with the exponential function as Y, the normal distribution on the X-axis and ...
4
votes
3answers
358 views

Plotting the convolution of the same probability density function with itself

I would like to compute the integral $$q_n(x)=\int_{B}^A q_{n-1} (\omega)f(x-\omega)\mathrm{d} \omega,\quad q_1=f,\quad n\geq 1.$$ for a given density function $f$. Basically it is the convolution ...
4
votes
2answers
142 views

what is The Formula behind the Kolmogorov-Smirnov test statistics?

I calculate the Kolmogrov-samirnov test statistic for the following data: ...
4
votes
1answer
147 views

Combinations which do not have elements in common

I can choose 2 letters from the four letters $\{A,B,C,D\}$ in 6 combinations using the combination formula $$\frac{n!}{ r! (n-r)!}$$ ...
4
votes
2answers
232 views

Function given exact arguments returns hugely different value than it returns when given equivalent inexact arguments

I was trying to compute the probability that a coin is from a particular underlying distribution given that a particular set of tosses was observed. (I know this can be done in a different way, but I ...
4
votes
4answers
134 views

How do I emulate DiscreteUniformDistribution in Mathematica?

How do I define a function from scratch that emulates DiscreteUniformDistribution? As shown below, attempting to use ...