Use this tag only if your question is 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 ...

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Probability of non-linear transformation

I'm reading about the accept-reject algorithm to generate non-uniform random numbers from the uniform. Let $X$ have a density on $\mathbb{R}^d$, and let $U$ be independent uniform on $[0,1]$. Then ...
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0answers
29 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with ...
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0answers
17 views

Intuition behind Mutual independence of sub-$\sigma$-algebras definition.

I was reading about Independence of sub-$\sigma$-algebras when I found the next definition: Let $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ $n$ sub-$\sigma$-algebras of $\mathcal{A},$ let $H$ be a ...
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63 views
+100

Exploiting the Markov property

I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs. Let ...
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1answer
50 views

Let X and Y be independent random variables such that $E|X+Y|<\infty$. Is it true that $E|X|<\infty$? Give a proof or a counterexample.

Let X and Y be independent random variables such that $E|X+Y|<\infty$. Is it true that $E|X|<\infty$? Give a proof or a counterexample. Thoughts: My intuition was to apply Fubini-Tonelli here ...
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0answers
17 views

Difference modes of convergence of a sequence of independent Bernoulli random variables

Suppose $(r_n)_{n \geq 1}$ is a sequence in $(0,1]$, $(X_n)_{n \geq 1}$ is a sequence of independent Bernoulli random variables such that: $P(X_n=0) = 1 - r_n, P(X_n = \frac{1}{r_n}) = r_n$. Show ...
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1answer
24 views

Convergence a.s. and convergence in $L^1$ don't imply each other [closed]

I'm trying to get two examples that convergence a.s. and convergence in $L^1$ don't imply each other. Now, I only know the examples that convergence a.s can't implied by convergence in probability, ...
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0answers
32 views

Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
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0answers
21 views

Minimum Random Variables and Integration

We are given a sequence of independent random variables $\lbrace X_{nk} \rbrace$, for $k=1,...,r_{n}$, with $E(X_{nk})=0$ and $\sigma^{2}_{nk}<\infty$. My question involves a small piece of the ...
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0answers
15 views

Kullback-Leibler Divergence (KL) and Approximation Symmetry Property

The Kullback-Leibler Divergence doesn't satisfy the symmetric property. But, it can be approximated (bounded) to such a value. in this paper: Compressing Interactive Communication under product ...
3
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1answer
55 views

Tic Tac Toe: What is the probability that a random player draws against an infallible player?

I have simulated a tournament between an infallible Tic Tac Toe player and one that chooses its moves randomly. Even after 5 million games, the infallible player wins every single game. I know that ...
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0answers
37 views

Difference of dependent central Chi-Square random variables with 2 degrees of freedom

Suppose we have $X$ and $Y$, both are dependent and complex Gaussian random variables with zero means and the same variance $\sigma^2$. The real and imaginary parts of $X$ and $Y$ are independent, ...
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0answers
31 views

Problems deriving probability generating function for the negative binomial distribution [closed]

My problem is the following: Part A.a I can't get the moment generating function to be what it states in the exercise. And I found other people asking the same question but they get a different ...
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0answers
7 views

optimization technique to find the best result

i have two outcomes from two types of test. both the results are not 100% accurate. Is there any technique available to extract the final result from these two outcomes?
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1answer
23 views

Inequality for Expected Value of a Convex Function

Let $X$ be random variable with $-1 < X < X_\max$ and $X_{\max} >0$ and take $a = \min\{1, \frac{1}{X_{max} }\}$, I was wondering is the following inequality hold $$ E\left[ {\frac{X}{{1 ...
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1answer
17 views

Let W and Z be two random variables such that W ≤ Z. Show that for any ε > 0, P(W > ε) ≤ P(Z > ε).

I feel like I should be able to use Markov's Inequality, but have not found an effective way to use it for this problem. Any help would be appreciated.
2
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1answer
36 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
11
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3answers
348 views

Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n ...
1
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1answer
62 views

Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
0
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1answer
13 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
4
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1answer
64 views

$\lim_{n \to\infty} E(|S_n|)= \infty$ for $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1, E(X_1)=0$

Problem: For $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1$ and $E(X_1)=0$ and $S_n$ denoting the partial sum of the RVs we have $$\lim_{n \to \infty} E(|S_n|)=\infty $$ My Approach: I ...
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0answers
28 views

What is the meaning of probability of an edge connected by two nodes in a graph

I am studying random graph models. While studying random graph models if we want to generate for instance erdos renyi's random graph model then we will have to place $n$ vertices and connect each pair ...
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0answers
48 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = ...
2
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0answers
52 views

How to Find Expected Value from a Joint Distribution?

I am trying to solve the following problem: Let $X$ be a random variable from a contaminated normal distribution. That is, let $B ∼\text{Bernoulli}(p).$ Then $X|B = 0 ∼ N(µ, τ^2 )$ and $X|B = 1 ...
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0answers
31 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
1
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1answer
27 views

Equality of Measures and Intuition

Caveat: I have no formal training in measure theory and am learning as I go. The concept in this question is puzzling me: Equivalent measures if integral of $C_b$ functions is equal I'll re-state ...
1
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1answer
38 views

Random Walk on $\mathbb{Z}$

So I thought I'd try to find the expectation of $\frac{X_n}{n}$. Let $q=1-p$ I did this by conditioning on the first jump using the law of total of probability with expectations. Before that ...
0
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0answers
23 views

Probability of most frequent x substrings

I have a string of length N and I have found x most frequently occuring (distinct) substrings of length L (with condition than they occur atleast thrice in the string). Size of alphabet is 4. I am ...
0
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0answers
59 views

Generalization of classic 3 roll die game to $n$ rolls

I am trying to generalize the following well-known 3 roll die problem: "We roll a single die no more than 3 times. We can stop immediately after the first roll, immediately after the second roll, or ...
2
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2answers
33 views

Ratio of Expected values of Boys to Girls

In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of ...
3
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0answers
26 views

Does a analytic joint distribution necessarily have continuous marginals?

Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely: Suppose $X$ and $Y$ are two ...
3
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1answer
41 views

Characteristic functions of random variables are non-negative definite

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X:\Omega\to\mathbb{R}$ a random variable. How to prove that the characteristic function $\varphi_X(t) = E[e^{itX}] ...
3
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1answer
42 views

Characterization of Normal RVs by uni variate version?

If $X$ is a symmetric $n$-dimensional random vector with mean $0$ then is it true that: \begin{align*} & X \text{ follows a multivariate normal law} \\ & \text{iff} \\ & \|X\| \text{is a ...
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1answer
38 views

Counterexample that a measurable function does not guarantee almost sure convergence

I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$ ...
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1answer
24 views

Characteristic function of Laplace distribution

I'm trying to derive the characteristic function for the Laplace distribution with density $$\frac{1}{2}\exp\{-|x|\}$$ My attempt: $$\frac{1}{2}\int_{\Omega}e^{itx-|x|}\mathrm{d}x$$ ...
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0answers
22 views

Convergence in distribution to a constant implies convergence in probability to that constant.

Suppose that $(X_1,X_2,...)$ is a sequence of random variables and that the distribution of $X_n$ converges to the distribution of the constant $c$ as $n\to\infty$. Then $X_n\to c$ as $n\to\infty$ in ...
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0answers
26 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
2
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1answer
53 views

If X, Y, Z are independent random variables, then X + Y, Z are independent random variables. [duplicate]

I found the same question (X,Y,Z are mutually independent random variables. Is X and Y+Z independent? here), but the answer uses characteristic functions and fourier inversion theorem, but this is ...
0
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0answers
31 views

Prove that ($Y_n$,$F_n$) is martingale. What does ($Y_n$,$F_n$) mean in this problem?

$X_1$, .... , $X_n$ are independent variables, $P(X_i=1)=p$, $P(X_i=-1)=q$, where $0<p<1$. $F_n=\sigma(X_1,.....,X_n)$ and $Y_n=(\frac{q}{p})^{X_1+....+X_n}$. The task is to prove that ...
6
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0answers
55 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
0
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1answer
32 views

Does this set of infinite binary sequences have positive probability?

The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the ...
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0answers
26 views

The series $\sum_{i=1}^n \frac{Y_i}{n} _{n\ge 1}$ converges? Where $Y_i = min(1,X_i)$ and $X_i$ is a serie of random variables. [closed]

I've been finding some problems to solve this kind of question when studying to my probability's exam. We should probably use the law of great numbers, but I can't see how. The complete exercice: Let ...
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0answers
27 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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1answer
28 views

Lévy-process property

I get a problem that comes up in the construction of the Lévy-Itõ decomposition. For a Lévy-process $X$ there is a independently scattered poisson random measure $N$, such that for each t, and for ...
2
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1answer
73 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
2
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0answers
33 views

Indicator Functions properties in Probability

This is probably quite a straightforward question but I just thought I should double check my working. Say for instance we have two fixed values $K_{1}$ and $K_{2}$ such that $K_{2} > K_{1}$ and ...
2
votes
1answer
19 views

Max Likelihood Examples, Stuck in Calculation [closed]

We get samples 2,4,8,16 be random instances that get from distribution with following PDF. maximum likelihood estimation of $ (\alpha, \sigma) $ is : $ \frac {2}{3 ln 2}, 2$. $ f_{\alpha, ...
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0answers
47 views

Lim Sup and Measurability of one Random Variable with respect to Another

Here, there is a common proposition in probability theory : Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that ...
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0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
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0answers
36 views

The probability density function of the product of independent exponentially distributed RVs

When $X$ ~ exponential distribution(10) and $Y$ ~ exponential distribution(15), and they are independent, find the probability density function of $Z = XY$ I just took an exam for probability ...