Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
109,312 questions
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
- 1,987
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Finding an exact closed (non-recursive) form formula for the probabilities in a game of Risk
Yesterday I enjoyed some rounds of RISK: Global Domination with a friend from university. It is a long-running in-joke that “True Random” is the cause of winning and losing certain battles.
Our ...
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Asymptotic equivalence of a binomial Sum :
I've been strugling to find an asymptotic sequence depending on both $\{ap\}$ and $a \in (0,1/2) $ of the following sum defined for all positive integers $p$ :
$
\sum_{k = 0}^{\lfloor ap \rfloor} \...
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Blind Spot Regarding the Correlation Coefficient
I've noticed that there is a strange unexplained thing about Pearson correlation coefficient
$$\rho(X,Y) = \frac{\operatorname{Cov} (X,Y)}{\sqrt {\operatorname{Var}X} \sqrt {\operatorname{Var}Y}
}$$
...
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Expected number of lattice points in a randomly inscribed triangle
Question. Let $C_R$ be the closed disk of radius $R$ centered at the origin. Let $T$ be a random triangle formed by three vertices $V_1, V_2, V_3$ chosen independently and uniformly from the ...
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Covariance of Unbiased Sample Variance Estimators with Overlapping Samples
Unbiased Variance Estimator
Let $x_1 , \ldots, x_N$ be iid sampled from X.
Let Y(N) denote the N-mean estimator given by
$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$
Let v(N) denote the unbiased N-...
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Upper envelope of a family of binomial–tail functions with linear penalties
Let $n,m$ be positive integers with $1 \le m < n$.
For each pair of integers $z,r$ with $1 \le z \le n$ and $0 \le r \le z$,
define the function
$$
u_{z,r}(x)
:= \Pr[\operatorname{Bin}(n-z,\,x)\...
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Conditioning that a random variable is another random variable
I was just solving problems on conditional probabilities. When I was taught this concept, most of the conditions were given for specific value. But then I saw an example where the condition was that X ...
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Is this a probability density function? [closed]
Given $fx(x) = \{ \frac{1}{\pi} \; \text{for} \; x_1 + x_2 \le 1$
I am required to state if the function represents a density function and prove why. I know that to prove it I must check that $f(x) \...
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When is the weak limit of indicator/characteristic functions an indicator/characteristic function?
I've been reading the following posts (Link1, Link2) about weak limits of indicator/characteristic functions. It is clear to me that in general the weak limit of indicator functions may not be an ...
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Coin Toss and Independence
In a lecture note, following is stated:
Consider the following two events. There lies in front of you a fair coin. Alice tosses it. Then Bob
tosses the same coin. Let $A$
be the event that Alice gets ...
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What is wrong with my calculation? (Inclusion-Exclusion-Principle)
Consider the alphabet $\mathcal A:=\{a,b\}$ and a uniformly drawn word with three letters $X:\Omega\to\mathcal A^3$ (note that from this we already have that the marginal distributions must be ...
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Given that $a_n \sim \operatorname{Unif}[0,n]$, calculate $\mathbb{P}(a_1 < \cdots < a_5)$
Let $a_n \sim \operatorname{Unif}[0,n]$ be a sequence independent uniform random variables. The goal of the problem is to find the probability that the first 5 of these random variables turn out to be ...
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How does the Black-Scholes equation handle drift? [duplicate]
I've been reading about the Black-Scholes formula on wikipedia and investopedia and it seems like a lot. My understanding is very limited, but naively it makes sense to me one would be able to assign ...
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Sequences of Independent Probabilities Intuition
I am working on a fairly simple problem:
"When rolling a fair dice 12 times, what is the probability of getting 2 of each number"
My immediate instinct was to calculate as follows, in this ...
