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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Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies $E(f(X_n)) \to f(0)$.

The following is problem 4 from Section 4.2 of "A Course in Probability Theory" by Kai Lai Chung. Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies ...
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18 views

Conditional Expectation with independent sub-sigma fields

Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We ...
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Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$. I can't find a proof on the wikipedia article, or if it's ...
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How to derive this new Poisson to characteristic function

i rewrite Poisson depending on p and q, but further derive with maple, takes a very long time to calculate, i am afraid that it can not be calculated, i guess the key is using substitution what is ...
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18 views

A basic doubt on independence of events in probability

Let $X_1$ and $X_2$ be i.i.d random variable. Now, in a book I see the following steps to calculate $P(X_1 < X_2 < x)$ $P(X_1 < X_2 < x)$ = $P(X_1 < x, X_2 < x, X_1 < X_2)$ ...
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27 views

A basic probability doubt on independence

Let $X_1$ and $X_2$ be two i.i.d continuous random variable. I need to find the probability that $P(X_1 < X_2)$. I know how to formally find the probability by integrating over appropriate region ...
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26 views

Please show me that with which formula, I can calculate pooled variance for unequal population variance?

When equal population variances, I can calculate pooled variance (as like part-b) But when unequal population variances, how to calculate pooled variance ( as like part-e)(also I underlined it ...
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sampling schemes for binomial distribution

Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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35 views

Random Walk probability

I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
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24 views

Convergence in distribution and convergence of expectation.

Let $\{X_n\}_{n\geq1}$ and $\{Y_n\}_{n\geq1}$ be two sequences of uniformly integrable iid random variables with distributions $F_n(x)$ and $G_n(x)$, respectively. If $$|F_n(x)-G_n(x)|\leq ...
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33 views

A generalization of the conditional expectation to kernels

Let $\left(\Omega_1,\mathcal{A}_1, P\right)$ be a probability space, let $\left(\Omega_2,\mathcal{A}_2\right)$ be a measurable space and let ...
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Trying to understand formula for the Survival Function (survival analysis) [migrated]

I'm trying to learn the Cox Proportional Hazards Model on my own, and found this link that describes it in clear terms. But when I get to formula (5) (S(t) = exp(−H(t))) I can't figure out where ...
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37 views

Probability on product spaces

I am having some trouble, more of an argument with someone else, about a simple question regarding product spaces. Let $X_1,X_2,\dots,X_n$ a set of independent and identically distributed random ...
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20 views

(simple) Expectation of random variable as a multipart function

Let the random variable $Y \in [0, \infty)$, a real number $\theta >0$, and the random variable $X$ such that $X = \theta - \min(\theta,Y)$, thus, $X \in [0, \theta]$. That is, $X = 0$ if $Y > ...
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34 views

Can it be confirmed in this state when state transition probability >= 25%

if have a 2 * 2 state transition matrix, can it be said that if one of cell probability >= 25% then it is confirmed in this state if it is < 25%, then it is not in this transition state if this ...
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23 views

Probability Space and Conditions

maybe a bit of a basic question but I don't see how to formally solve it right now. In order to analyze a stochastic process, I want to change my probability space to a more convenient one (we can ...
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37 views

Order of convergence of a sum

Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that \begin{align*} \mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty. \end{align*} ...
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39 views

Help me solve the invariant measure of $Q$

My $Q$ matrix is given by: \begin{bmatrix} -\lambda &0 &\lambda &0 &0 &... \\ \mu&-(\lambda+\mu) &0 &\lambda &0 &... \\ 0&\mu ...
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33 views

On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...
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24 views

Is there a conditional version of the asymptotic equipartition property?

Let $X_i$ be independent random variables with $\operatorname{Pr}(X_i = x) = p_x$, and let $F_n$ be the empirical frequency distribution of $X_1, \ldots, X_n$: that is, $(F_n)_x$ For any frequency ...
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22 views

Proof that the radius is a sufficient statistic for a circle

How can I prove that the radius of a circle is the sufficient statistic for the probability of choosing random points in the area of the cricle?
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25 views

Using the Chebyshev Inequality

This is the Q: A 20 fair coins tosses, (f means the "H" of the coin). I have to block the probability that I will get n/2+n/100 "H"-s by Chebyshev Inequality. [n=20 in this case...], so: n/2+n/100 = ...
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51 views

Stochastic process, Gaussian, with zero mean is a Wiener process

Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
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23 views

Continuity of a certain matrix-like function

Let $X$ be a finite set and let $M$ be a space of all probability measures on $X$. Let $f:X\to\Bbb R^{m\times m}$ be a random matrix and consider a function $c:M\to\Bbb R$ defined as $$ c(\mu) := ...
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25 views

Ratio of PDF to complementary CDF

Let $f(x)$ be a probability density function, and $F(x)$ be the cumulative distribution function of $f(x)$. $$F(x) = \int_{-\infty}^{x}f(u)du$$ Then intuitively, what does the following ratio ...
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43 views

Find the limiting distribution

Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
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57 views

Proving that $P\left(\bigcup_{j=1}^n E_j^{(n)}\right)\sim\sum_{j=1}^n P(E_j^{(n)})$ for independent events $E_j^{(n)}$

For arbitrary events $\{E_j, 1\le j\le n\}$, we have $$P\left(\bigcup_{j=1}^n E_j\right)\ge\sum_{j=1}^n P(E_j) - \sum_{1\le j < k \le n} P(E_jE_k)$$ If $\forall n: \{E_j^{(n)}, 1\le ...
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Does a product measure on a product space constructed from two sub-fields of the same space determine a measure on the underlying space?

Let $\mathcal{A}_1,\mathcal{A}_2$ be $\sigma$-algebras on $\Omega$. Let $P$ be a probability on $\mathcal{A}_1$ and let $Q$ be a Markov kernel from $\mathcal{A}_1$ to $\mathcal{A}_2$. Set $K:=P\otimes ...
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$E(|X+Y|^p)\ge E(|X|^p)$

If $X$ and $Y$ are independent, $E(|X|^p)<\infty$ for some $p\ge 1$, and $E(Y) = 0$, then $E(|X+Y|^p)\ge E(|X|^p)$. Is it maybe true that for each fixed $x$ that the following inequality is true? ...
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Levy process absolute moment

For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
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47 views

Probability Joint PDF

Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf: f(x) = Ax^2 for 0 < x < 10 where A is a constant that you need ...
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Calculating the probabilities of different lengths of repetitions of numbers of length 6

This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4 except now I'm having problem with numbers of length 6. ...
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A basic doubt on CDF and pdf

Let $X$ be a random variable and $A$ be a Borel subset of $R$. Then $$P(X \in A) = \int _{A}f_X(x) dx$$ where $f_X$ is the probability density function of the random variable. Now I have not ...
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If $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$.

If $X$ and $Y$ are independent and for some $p>0$: $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$. How can I go from $E(|X+Y|^p)<\infty$ using independence to ...
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B.F.'s generated by disjoint subfamilies are independent

Problem 5 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung If $\{X_\alpha\}$ is a family of independent r.v.'s, then the B.F.'s generated by disjoint subfamilies are independent. ...
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Analytic tools in the theory of Galton-Watson processes

The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
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32 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
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If the fields $F_\alpha^0$ are independent, then so are the B.F.'s $F_\alpha$.

Problem 4 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung Fields or B.F.'s $F_\alpha(\subset F)$ of any family are said to be independent iff any collection of events, one from ...
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41 views

On unions of independent events

If the events $\{ E_{\alpha}, \alpha\in A\}$ are independent, then so are the events $\{F_\alpha,\alpha\in A\}$, where each $F_\alpha$ may be $E_\alpha$ or $E_\alpha^c$; also if $\{A_\beta, \beta\in B ...
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37 views

Orthogonalization of a set of random vectors

Suppose $w_1$ and $w_2$ are zero-mean jointly Gaussian random vectors. Further suppose that they have a covariance matrix given by $$ \mathbf{cov}\begin{bmatrix}w_1 \\ w_2\end{bmatrix} = ...
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Expectation of function of stochast

I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
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Martingale and indicator

Exercise comes from "1000 exercices in probability" (12.9.6). Let $X_1, X_2, \dots$ be independent random variables with $X_n=\begin{cases} 1, & \text{with probability} & (2n)^{-1}, \\ 0, ...
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How does this violate probability theory?

Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$) $p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$ $p(Y = -1) = .5$, $p(Y = 3) = .5$ Question: Despite not being handed any information ...
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Uniform distribution on the n-sphere.

I have the next RV: $$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$ where $$X_i \tilde \ N(0,1)$$ It's a random vector, and I want to show that it has a uniform ...
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Joint distribution of multiple binomial distributions

In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them. The original file can be ...
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32 views

Fisher information of a Binomial distribution

The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
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68 views

Law of large numbers?

Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$: If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
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36 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
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31 views

Approximation of a random variable by a sequence of simple random variables

It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
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29 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...

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