Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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690 views

probability for a $n\times n$ matrix to have no complex eigenvalues

Let $A$ be a $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has no complex eigenvalues? The answer cannot be 0 or 1, since ...
255 views

Probability of a group being finite

Suppose $F_m := F[x_1, … , x_m]$ is a free group on $m$ generators $x_1, … , x_m$ and lets define Cayley ball $B_m^n := \{e, x_1, x_1^{-1}, … , x_m, x_m^{-1}\}^n$ as the set of all elements with ...
746 views

Random sum in coupon collection

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
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Generating function for number of $r$-disjoint subsets each of size $k$

Fix $n, k$. Then $$C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!}$$ is the number of ways to form $r$ disjoint subsets each of ...
280 views

Randomly Generate Probability Mass Function With Specific Entropy

How can I randomly generate a probability mass function such that the entropy of a random variable that follows that probability mass function is a specific value $h$? Basically, I need to randomly ...
900 views

Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random Variable)...
408 views

Extracting an (almost) independent large subset from a pairwise independent set of Bernoulli variables

Let $n>1$, and let $X_1,X_2, \ldots ,X_n$ be non-constant random variables with values in $\lbrace 0,1 \rbrace$. Let us say that a subset of variables $X_{i_1},X_{i_2}, \ldots,X_{i_d}$ is complete ...
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Asymptotic value of card drawing game

A deck consisting of $r_0$ red cards and $b_0$ black cards is randomly shuffled. The host turns up the cards one at a time; if it is red, you get $\$1$; otherwise you pay the host$\$1$ (and you're ...
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Is kernel density estimation a GMM with uniform mixture weight?

recall that for a Gaussian Mixture Model, the density of p(x) (multivariate) is $$P(x) = \Sigma_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ On the other hand, non-parametric density estimation ...
892 views

How do Kolmogorov 0-1 law and CLT imply normalized sample mean doesn't converge in probability nor a.s.?

From WIkipedia the central limit theorem states that the sums Sn scaled by the factor $1/\sqrt{n}$ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-...
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Random graph connectivity, and the existence of isolated vertices

Here $G_{n,p}$ represents the Erdős-Rényi random graph model, where the graph has order $n$ and each edge is added independently with probability $p$. I am faced with proving the following claim: ...
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Betting game with numbers

Someone is challenged to play the following game: there are 36 marbles in an urn. 21 of them are red, 9 are blue and 6 are green. Each red is worth 0 rupees, each blue 100 rupees and each green 1000 ...
241 views

Is this an okay way to calculate covid-19 death rates?

I'm a tad rusty on my math skills but this isn't too hard. Can you confirm that I am doing this right. I wanted to know what percent of people die from covid-19 so I started looking for statistics. ...
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Flies in a cube

Two flies sit in the corners $(1, 1, 1)$ and $(n, n, n)$ of a discrete cube $M^3$, where $M=\{1,\ldots, n\}$. After every unit of time both flies decide randomly and independently of each other ...
678 views

Accounting for uncertainty in an Elo rating system for Foosball

For a Foosball game at work we implemented a rating system based on the Elo system. Allthough we achieved a sensible result so far, which provided us with a lot of fun (which is the goal) we feel we ...
222 views

What is the probability that the Golden State Warriors will break the NBA regular season record of wins?

There are $82$ games in a regular season, and the current record is held by the Chicago Bulls, at 72-10. As of yesterday (March 4th 2016), the GSW season performance stood at 55-5. Assuming they ...
251 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation -\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot \big(G(y)...
163 views

Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
According to wikipedia: Assume that $X_1,X_2,\dots$ are independent and identically-distributed random variables in $\mathbb{R}$ with common cumulative distribution function $F(x)$. The empirical ...