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Results tagged with probability-distributions
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user 2954
In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.
5
votes
Accepted
Minimum probability that two Gaussian random variables are small
The minimum value is simply $2\alpha-1 = 0.365379$ where $\alpha = \Phi(1)-\Phi(-1) = P(|X|<1)$ where $X \sim N(0,1)$. This can be achieved by translating the percentile of $X$ (considering the percen …
4
votes
Does the optimal strategy converge in poker if the SPR tends to infinity?
The Clairvoyant Game
Here is a well-known toy problem (the Clairvoyant Game) that doesn't converge: Suppose your hand is face-up. You have no hidden information. You don't know whether your opponent' …
3
votes
Accepted
Should you bet in poker against Darth Vader?
Here is an answer to the updated question:
Suppose that there are two betting rounds. Darth Vader has three types of hands. Type 1 wins with probability 1. Type 2 is a draw that hits (becomes a winni …
6
votes
Should you bet in poker against Darth Vader?
Getting all-in while behind
It is not just when you are ahead that you might want to get all-in against someone who has an information advantage. Suppose the pot is $1$ and the effective stack depth …
1
vote
Comparison of tail behaviour of two (bounded) random variables given their moments
First, here is a counterexample to a simpler statement with $a_0$ fixed at $0$: Let $Y$ be the constant random variable $2$. Let $X$ be $1$ with probability $1/2$, and $3$ with probability $1/2$. Then …
2
votes
A generalization of negative binomial distribution
This is related to the coupon-collector problem. These random variables have been studied by many people, although I don't recall a particular name for them. See, for example, Anna Pósfai's thesis (ab …
3
votes
Accepted
divisibility of uniform distribution
Here is a direct argument.
Suppose independent $X_1,X_2 \sim X$, and $X_1+X_2$ is uniform on $[0,1]$.
$X$ is supported on $[0,1/2]$.
For any $0 \lt \alpha \lt 1/4$,
$\alpha = P\left(X_1+X_2 \in [ …
4
votes
Brownian motion, crossing intervals, possible usage of second moment method?
Here is some notation I used in the related problem with some results. Let the probability hat a Brownian motion starting at $0$ returns to home on $[a,b]$ be $h(a,b) = h(b/a)$. By the calculation in …
7
votes
Accepted
Number of intervals needed to cross, Brownian motion
I'll address the second question on the expected value of the sum $K_n$.
Let $\phi(x)$ and $\Phi(x)$ be the probability density function and cumulative distribution functions for a standard normal di …
4
votes
Accepted
Are all mixtures of these unimodal functions unimodal?
Here is a counterexample:
$$f(y) =\frac{1}{10}\frac{y^{3/2}}{(y+1)^2} + \frac{9}{10}\frac{y^{3/2}}{(y+10)(y+1000)} $$
This is not unimodal because it takes the values $14281/439250 = 0.0325$ at $y= …
2
votes
Random partitions with prescribed pairwise membership probabilities
There are many more partitions than there are pairs for large enough $n$, so rank-nullity says uniqueness will fail. $\frac12 (12) + \frac12 (34) \sim \frac12 (12)(34) + \frac12 e$ where we identify …
5
votes
Does bounding moments make distributions close in total variation distance?
No, you need some other condition if you want to restrict the total variation distance. The total variation distance is maximal between a discrete random variable and a continuous one, and you can tak …
2
votes
Random weighted selection without replacement
This model has come up on MO before: Drawing natural numbers without replacement. I don't think it has a standard name in mathematics.
In mathematical analysis of poker, this is called the Independen …
2
votes
How do you call the problem of approximating a continuous distribution with a simple discret...
It sounds like you want evenly spaced quantiles of a distribution. If you have to represent a distribution with one number, the median may be a reasonable choice. If $n=9$, then you can use the $10$th …
7
votes
Accepted
The average number of people that can sit on a bench of a given length.
Assume that the process stops when someone can't fit.
I believe the distribution of the amount of overshoot is known as a ladder height distribution, and that this is in Feller's classic text, but I …