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Results tagged with probability
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user 270937
A probability provides a quantitative description of the likely occurrence of a particular event.
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What is the 'same distribution' mean?
Strictly speaking, it means that the CDF is the same.
That is, the type of distribution, the mean, the variance, and all parameters are all the same, if they are well-defined.
For most of the commonly …
- 185
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Can two different measures have the same first order stochastic dominance?
$(S,\Sigma,\mu)$ is the common probability triple, where $S=[0,1]$, $\Sigma$ is the Borel sigma algebra, and $\mu$ is the Lebesgue measure.
$X:[0,1]\to\mathbb R$ is a r.v. …
- 185
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Can two different measures have the same first order stochastic dominance?
Assume that $\mu, \nu$ satisfy $X \: \mu\text{-FOSD} \: Y \Longleftrightarrow X \: \nu\text{-FOSD} \: Y$ for all X, Y.
For $A, B \subset S$, such that $\mu(A) < \mu(B)$ denote
$$X := 2\chi_{\bar A}, …
- 185
1
vote
0
answers
57
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First order stochastic ordering implies countably additive probability measure?
Let $P$ be a finitely additive probability measure. I learn from my friend that:
$[P(X>Y)=1 \implies \mathbb E_P[X]>\mathbb E_P[Y]]\iff$ $P$ is countably additive.
Seems to be a very useful result. …
- 185
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2
answers
202
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How to understand the definition of Markov Chain $P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B)$?
The definition of Markov Chain in Durrett (Probability: Theory and Examples, 2019, Section 5.2) is:
$$P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B), $$
where $p$ is the Markov transition kernel distribution …
- 185