Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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67 views

Formula to fit a straight line to data

Theorem (Best Linear Prediction of $Y$ outcomes): Let $(X,Y)$ have moments of at least the second order, and let $Y'=a+bX$. Then the choices of $a$ and $b$ that minimize $Ed^2(Y,Y')=E(Y-(a+bX))^2$ ...
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2answers
51 views

question about joint and marginal distribution

I have a question about how to compute the joint and marginal distribution. Let $x$ and $y$ have the Gaussian densities. $x \sim \mathcal{N}(m, P), \ y|x \sim \mathcal{N}(Hx, R)$ how to compute ...
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1answer
365 views

Infinite Coins Tossed Infinitely Often [duplicate]

If an infinite number of coins are tossed infinitely often, is it true that there will be infinite subsets of those coins that repeat any finite sequence of heads/tails infinitely often? I.e., ...
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1answer
66 views

How can I compute Expectation of Arbitrary Expressions over Random Variables?

Mathematica seems to be doing it: http://www.wolfram.com/mathematica/new-in-8/probability-and-statistics-solvers-and-properties/compute-the-expectation-of-an-expression.html I'm interested in how can ...
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1answer
36 views

If $f$ is any function and $X_1 … X_n$ are IID, are $f(X_1), f(X_2), …, f(X_n)$ IID?

Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ be any function and let $X_1, X_2, ..., X_n$ be IID real-valued random variables drawn from any arbitrary distribution. Is it guaranteed that ...
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1answer
138 views

Counterexample for finite dimensional weak convergence

Could you give an explicit construction of a sequence $\mathbb P_n$ of probability measures on $C[0,\infty]$ which converges in the sense of finite dimensional distributions BUT does not converge ...
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1answer
91 views

A basic combinatorics/probability question

Suppose you have a set $A = \{1, 2, 3, 4, 5, 6, 7\}$. You do a sampling with replacement over A (uniformly at random), and get a multiset/bag $B$ of size $5$. For instance, B could be $\{2, 3, 4, 5, ...
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0answers
282 views

Variance of minimum of N random variables

Let $X_1,X_2,\dots,X_N$ be i.i.d. random variables with support $[0,M]$, and with density and distribution functions $f_X(x)$ and $F_X(x)$ respectively. Given $Y= \min_{i\in \{1,\dots,N\}} X_i$ , ...
3
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1answer
86 views

Is it possible to sample the Dirac delta function?

The Dirac delta function can be a probability measure with the unit/Heaviside step function as its cumulative distribution function. Is it possible to sample such a distribution? If a random variable ...
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1answer
101 views

Can someone check if this proof of the strong law of large numbers is correct?

I think I have a proof of the strong law that is slightly different than some of the others I have seen. I would appreciate another eye to see if it makes sense. I am assuming the following ...
2
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1answer
100 views

Computation of a mean (random sum)

Let $X_1$, $X_2$, ... be independent and identially distributed positive random variables and define the sum $S_n = X_1 + X_2 + ... + X_n$. Consider the first time $N$ where $S_N \ge b$ with a given ...
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1answer
61 views

Lebesgue density for other probability measures on $[0,1]$

Does the Lebesgue density theorem hold for arbitrary (Borel) probability measures on $[0,1]$? Following Downey & Hirschfeldt's proof leads me to believe that the answer is "yes". (Recall every ...
3
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1answer
123 views

equivalent condition for moment generating function

Consider a random variable $x$ with pdf $f(x)$, and have $x \ge 0$. The moment generating function is defined as $M(t)=\int^{\infty}_{-\infty}e^{-tx}f(x)dx$ (noted that we change the sign of $t$ ...
2
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1answer
297 views

Wald's second equation

We have a random walk $S_N=\sum_{i=1}^{N}X_i$ where $X_i$ are i.i.d with $0<E(X_i)<\infty$ and $N$ is a stopping time. What is the "exact" second equation of Wald ? I've seen different results ...
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43 views

Estimate a level set of the form $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$

Suppose I have a continuous function $f(\mathbf{x}):\mathbb{R}^d \mapsto\mathbb{R}$. I am interested in the level set $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$. Suppose the lebesgue measure ...
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1answer
33 views

Convergence in probability of the sum of scheme of series

Could you please help with this one. It looks like smth simple but I can't figure it out. Let $\{x_{in}\}, \ i=1,\dots, n, \ n=1,\dots,\infty$ be the scheme of series of random variables. For each ...
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2answers
431 views

Almost sure convergence of stochastic process

Suppose that we have a (almost surely) continuous stochastic process $\{ X_{t} \}_{t \geq 0}$ on $[0,1]$ with non-stochastic initial value $X_{0} = x_{0} \in [0,1]$ and exponentially decreasing ...
3
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2answers
291 views

Blackwell–Girshick equation?

We have the following theorem: Let $N$ be a random variable assuming positive integer values $1, 2, 3,\dots\,$. Let $(X_i)$ be a sequence of independent random variables which are also independent of ...
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1answer
1k views

How to prove if P(A|B)>P(A) then P(B|A)>P(B) [closed]

How to prove that If P(A|B)>P(A) then P(B|A)>P(B)
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41 views

$Var(X-Y)$ strange outcome

THe time $Y$ between the arrival of two jobs has the probabilitiy density $$ f(y) = \frac{1}{2}e^{-\dfrac{y}{2}} $$ The duration, $Z \sim Unif[1,3]$ and is independent of Y Assume that ...
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1answer
104 views

Convergence in probability, continuity and uniform convergence in probability

Let $(X_i)_{i\in\mathbb{N}}$ be a strictly stationary sequences of real valued random variables with finite variance. We have the empirical distribution functions $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n ...
3
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1answer
66 views

Convergence in distribution and normality of the limit

Let $Z=(Z_1,Z_2)$ be a bivariate standard normal vector and $Y_{1,n},Y_{2,n}$ two sequences of real valued random variables with finite variance such that $Y_{1,n}\xrightarrow{d}Z_1$ and ...
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1answer
110 views

What is meant by “constant” in the optional stopping theorem?

One of the three condition of the optional stopping theorem is that "There exists a constant $c$ such that $|X_{t\wedge \tau}| \leq c$ a.s. for all $t\in \mathbb N_0$". In the article of Wikipedia on ...
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1answer
67 views

Feller Processes and the resolvent

To get you on the same page as I, I am following "Continuous time Markov Processes" by Thomas Liggett, and referring to Chapter 3. In order to make the question self-contained, I will mention that ...
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0answers
47 views

From one nonexplosive markov chain to another

Fix a continuous time Markov Chain. (I am using Liggett's definition in Ch.2 which is that paths have to be right continuous with left limits and the state space is countable. Everything else is ...
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1answer
63 views

Prove average converges with diminishing $\epsilon$

How can one prove this claim? Seems like neither Chernoff nor Hoeffding bounds work. Chernoff bound is not additive. And since it must be that $\epsilon$<$1/n$, Hoeffding bound will not go to zero. ...
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1answer
259 views

Expectation of Truncated distribution with two random variables in conditional

How to find the conditional expectation $E[A_1|A_1\ge A_m,A_2 \ge A_m,A_1+A_2 \ge 2A_y]$ where $0 \le A_1 \le 1$; $0 \le A_2 \le 1$; $\frac{1}{2} < A_m < A_y < 1$; $A_m, A_y$ are constants; ...
3
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0answers
45 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
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1answer
461 views

What is the Expectations of all 3 ants meeting at same point?

Say we have 3 ants in three corner's of triangle. What is the expectations that all 3 ants meeting together given that the ant moves in any direction. So by just seeing it I figured out that in 2 ...
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1answer
61 views

Expectation of a conditional expectation with more than one variable in conditional part

Suppose we have three indicator variables: $I_1 =1$ or $0$ with some probability $I_2 =1$ or $0$ with some probability $J=1$ or $0$ and the value of $J$ depends on $I_1, I_2$. $I_1, I_2$ are ...
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69 views

Conditional expectation of two random variables

Let A$_1, A_2$ be two i.i.d random variables such that $0< A_1 <1$ and $0< A_2 <1$. Let $A_m$ be a constant such that $0< A_m <1$. Can we say that $\mathbb{E}[A_1 | A_1 < A_m, ...
3
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2answers
191 views

Variance of summation of Bernoulli variables

Let $X_1,\ldots,X_n$ be independent Bernoulli variables, with probability of success $p_i$ and let $Y_n =\frac1n\sum\limits^n_{i=1} (X_i - p_i )$ a) find the mean and variance of $Y_n$ b) ...
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5answers
153 views

why is $E[E[Y|X]] = E[Y]$

I have a derivation from my book, I have a problem with the very first line: $$ \begin{align} E[E(Y|X)] &= \int_{-\infty}^\infty E(Y|x)f_1(x)dx <- \text{why dx}\\ &= ...
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1answer
79 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
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2answers
66 views

What happens when a probability actually occurs?

Sorry if I am mixing up the model with the reality, but when for instance a low probability occurs, what happens with the rest of the probability? Philosophically I find it hard to argue that any ...
2
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1answer
232 views

Can someone explain the intuition behind this moment generating function identity?

If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= ...
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1answer
253 views

Uniform Law of large numbers

Could you please help with the proof of the proposition: Let $G(\cdot)$ be bounded, continuous, strictly increasing function on $\mathbb{R}.$ Let ${\xi_t},\, t\in\mathbb{Z}_+$ be i.i.d random ...
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0answers
39 views

Poisson distributed variable after iterative process

The value of $x$ is changed in a stochastic iterative process. Changes of $\pm1$ are possible. I am searching transition probabilities $p(x=n \rightarrow x=n+1)$ and $p(x=n \rightarrow x=n-1)$ that ...
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1answer
117 views

How to vary lambda in exponentially distributed numbers

I am implementing an exponentially distributed random number generator (RNG) based on George Marsaglia's Ziggurat algorithm. I previously used the algorithm to create a normally distributed RNG. By ...
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3answers
102 views

Conditional probability of IIDs

$\{X_i\}$ are i.i.d. standard Normal variables, with mean 0 and standard deviation 1, $$S={1\over{n}}\sum_{i=1}^nX_i$$ What is the conditional probability $P(X_1|S\ge1)$?
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1answer
155 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
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1answer
70 views

distribution of $\cos(\omega_0 n)$ where n are integers?

Assume we have the sequence $\,x[n]=\cos(\omega_0 n),$ where $n$ are integers. If we suppose these are realizations of a random variable, what would be the p.d.f. of that random variable?
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1answer
101 views

Equality of sets when minimizing Shannon's Entropy

Let $P = \{p_1, \ldots, p_n\}$ be a set of probabilities, i.e., $0 \leq p_i \leq 1$. $P$ is such that $\sum_{p_i \in P} p_i = 1$. I have a set of actions $\mathcal{A} = \{a_1, \ldots, a_N\}$ that can ...
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1answer
94 views

Why universally and not just Borel policies

In a famous book Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve they use universally measurable policies that come up with some handy features: e.g. they show that every ...
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2answers
56 views

Probability of getting green

If a bag contains x black balls, y green balls and z yellow balls , now a ball is drawn : 1. It's yellow -> try again 2. it's green -> stop 3. it's black -> stop What is the probability of getting ...
2
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2answers
176 views

Probability of getting white ball

If there are x black balls, y white balls and z red balls in a bag , r balls are drawn without replacement and now after this one more ball is drawn . What is the probability of this ball being white ...
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2answers
57 views

Optional sampling

Let $(X_i)_{i\in\mathbb{N}}$ be iid random variables with $\mathbb{E}|X_1|<\infty$ and let $S_n \stackrel{\rm{}def}{=} X_1+\cdots+X_n$ for all $n\in\mathbb{N}$. If $T$ is a stopping time with ...
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1answer
638 views

Going from the Poisson distribution to the Gaussian.

In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value ...
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1answer
51 views

approximating $\frac{S^2}{\sigma^2}$

Let $Y_1,\ldots,Y_n$ be independent random variables from a normal distribution with expected value $\mu$ and variance $\sigma^2$ and let $S^2 = \dfrac{1}{n-1} \sum^n_{i=1} (Y_i-\bar{Y})^2$ be the ...
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0answers
63 views

Iterative process that leads to Poisson distribution

I want $x$ to be Poisson distributed. I will call occupation probability $p(x=n) =: p(n)$ and the transition probability $p(x=n \rightarrow x=n+1) =: p(n \rightarrow n+1)$ The value of $x$ is ...