Questions tagged [probability-theory]

For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

The modern theory of probability is formulated using . Use this tag if your question either involves the theoretical foundations of probability or you are seeking responses at the level or rigor used in modern probability theory. Examples of subtopics include probability spaces, , convergence of random-variables, , , and other , applications of famous theorems including the strong and weak laws of large numbers, the central limit theorem, the law of the iterated logarithm, et cetera.

Use for explicit computation of probabilities or expectation values, and use for specific distribution functions.

44463 questions
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Zero probability and impossibility

I read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering if…
Tim
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Are functions of independent variables also independent?

It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$…
LLS
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Independence and conditional independence between random variables

For a family of random variables, I was wondering if independence and conditional independence under any condition among them imply each other? If not, can these two concepts imply one another under some special cases?
Tim
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31
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5 answers

Why isn't there a uniform probability distribution over the positive real numbers?

Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what probability distributions can exist over the positive…
Casebash
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27
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2 answers

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far can this phenomenon be pushed? The linked…
anon
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2 answers

How to formalize "conditional random variables"

I've been using "conditional random variables" as a notation aid with some good success in problem solving. But I've heard people claim that one shouldn't define conditional random variables. By a conditional random variable for $X$ given $Y$, a…
nomen
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23
votes
2 answers

Central Limit Theorem implies Law of Large Numbers?

Let $X_i$ be iid random variables and let $\overline{X}_n=(X_1+\cdots+X_n)/n$. If $EX_i=\mu$ and $\operatorname{Var}X_i = \sigma^2$ then the central limit theorem says that with some conditions we have convergence in some…
user782220
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23
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The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable

I'm trying to prove this conclusion but have some problems with one of the steps. Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian. We use characteristics function…
pluskid
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19
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5 answers

Can a Dirac delta function be a probability density function of a random variable?

Can the Dirac delta function (or distribution) be a probability density function of a random variable. To my knowledge, it seem to satisfy the conditions. To my interpretation getting a positive real number as the outcome is 1 and that for a…
Rajesh D
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1 answer

What is the difference between the probability law of X and the distribution of X?

I am a bit confused. The probability law regarding a random variable is defined as mapping $\mathcal{P} : \mathbb{B} \to [0,1]$, where $\mathbb{B}$ is the regular Borel set; that is, a probability law is an image measure. On the other hand, the…
user3503589
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Expected value is a linear operator? Under what conditions is median also a linear operator?

I have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this? Also, I know that generally median $Med()$ is not a linear…
cinny
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16
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1 answer

How to obtain tail bounds for a square of sub-Gaussian random variable?

A zero mean sub-Gaussian random variable $Z$ satisfies ${\mathbb E} \exp(tZ) \leq \exp(t^2\sigma^2/2)$ for some constant $\sigma > 0$. This bound can be used together with the Chernoff bound to obtain a two sided tail bound. I am interested in…
mkolar
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1 answer

From conditional probability to conditional expectation?

Conditional probability is defined in terms of conditional expectation as $$ P(A \mid \mathcal{N}) := E(I_A \mid \mathcal{N}) $$ where $(\Omega, \mathcal{F}, P)$ is a probability space, $\mathcal{N}$ is a sub $\sigma$-algebra of $\mathcal{F}$, and…
Tim
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14
votes
5 answers

When can a random variable be written as a sum of two iid random variables?

Suppose $X$ is a random variable; when do there exist two random variables $X',X''$, independent and identically distributed, such that $$X = X' + X''$$ My natural impulse here is to use Bochner's theorem but that did not seem to lead anywhere.…
robinson
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14
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2 answers

why distribution function is right continuous?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the exercise: 2.2 a) (iii). Let $X:\Omega \rightarrow \mathbb{R}$ be a randome variable. The distribution function $F$ of $X$ is defined by $$F(x) = P[X\leq x]$$ a)…
athos
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