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Questions tagged [probability-theory]

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 [tag:probability-...

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Proving Order Statistics Are Independent

I am working on the following problem: Let $X_1, X_2,...$ be independent random variables with a common continuous distribution function. Let $T^{(n)}(\omega) = (T_1^{(n)}(\omega),..., T_n^{(n)}(\...
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1answer
18 views

Equation for the expected value of a discrete random variable

I'm reading the book Introduction to Stochastic Processes, p.24. In proving the expected value for 'any non-negative random variable $X$', the author provides the following equation for the expected ...
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1answer
18 views

Showing Function is Measurable

I'm struggling with the following problem: Let $(X, \mathcal{X})$ and $(Y, \mathcal{Y})$ be measure spaces, and suppose that for each $x \in X$ there is a probability measure $\nu_x$ on $(Y, \mathcal{...
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14 views

Existence of a regular version of the conditional probability

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $\Omega$ $(E_i,\mathcal E_i)$ be a measurable space $Y:\Omega\to E_2$ be $(\...
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1answer
36 views

Find the distribution function and the density function of random variable $Y=X_1+X_2$

Working on a problem on probability and I just cannot get myself to make any progress. Supposed that the 2-dimensional random vector $X=(X_1, X_2)$ has joint p.d.f $f(x_1,x_2)$, i.e. $f$ is a non-...
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1answer
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Why can we substitute any arbitrary variable in for this expression of the CDF?

I'm trying to follow this example of a variable transformation, which defines the PDF and CDF as follows: PROBLEM: The above definition is used in solving this problem: Question: Now, where I get ...
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1answer
26 views

An empirical central limit theorem

I want to prove the following CLT: Let $(X_n)_{n}$ be i.i.d random variables with $\mathbb EX_1=0$ and $\mathbb E(X_1^2)<\infty$, then $$\frac{\sum_{i=1}^{n}X_i}{\sqrt{\sum_{i=1}^{n}X_i^2}} \...
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Convergence of i.i.d. random variables with respect to index

This is exercise 2.3.17 in Probabilty:Theory and Example p67. $X_1,X_2,X_3\cdots$ are i.i.d random variables. The reader is reqeusted to find the equivalent condition for each of the four statement. ...
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30 views

Permutation that Orders Random Variables is Uniformly Distributed

I am working on the following problem: Let $X_1, X_2,...$ be independent random variables with a common continuous distribution function. Let $T^{(n)}(\omega) = (T_1^{(n)}(\omega),..., T_n^{(n)}(\...
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Minimum Variance Unbiased Estimator (MVUE) is unique proof - wrong [on hold]

Here is the proof of "MVUE is unique" that my lecturer gave: Now I understand the following: The first expansion is done using the formula for the sum of correlated random variables (https://en....
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Verification: Use PGF to find mean and variance of discrete random variable $X$.

Let $X$ be a discrete random variable with PMF $\mathbb{P}(X = k) = \frac{1}{8}(7/8)^k$ for $k = 0,1,2,....$. Use PGF $G(s) = \sum_{k = 0}^\infty s^k\frac{1}{8}(7/8)^k$ find find the mean and ...
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1answer
19 views

Show some property of a Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I\subseteq\mathbb R$ $(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be ...
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1answer
26 views

Let $X_1, X_2$ be standard normal, Show $Y_1,Y_2$ are either independent or not independent.

$X_1$ and $X_2$ are standard normal. So they each have pdf's $\frac{1}{\sqrt{2\pi}}e^\frac{-x_1^2}{2}$ and $\frac{1}{\sqrt{2\pi}}e^\frac{-x_2^2}{2}$ Define $Y_1 = X_1+X_2$ and $Y_2 = (X_1 - X_2)^2$. ...
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1answer
20 views

Can $A$ and $B$ be independent of each other?

If event $A$ is dependent on event $R$ and so is even $B$, but there is no direct relation between $A$ and $B$. Can $A$ and $B$ be independent of each other? To give an example, suppose there is a $\...
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1answer
18 views

Mean of zero mean random variables has Cauchy-Lorentz distribution under constraints on the characteristic function

Take $X_1 , X_2 , \cdots $ are $iid$ with zero mean. Take $$Z_n = \frac{X_1 + \cdots X_n}{\sqrt{n}} \stackrel{d}{\rightarrow} X$$ and $$Z_{2n} = \frac{X_1 + \cdots X_{2n}}{\sqrt{2n}} \stackrel{d}{\...
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1answer
34 views

Probability Quadratic Has Real Roots

Suppose that $A, B, C$ are positive, independent random variables with distribution function $F$. I am trying to show that quadratic $Az^2 + Bz + C$ has real roots with probability $$\int_0^\infty \...
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25 views

Independent random variables $(X_i)$ have the same law, then $(X_i,\sum X_i ) $ have the same law for any $i$

Let $(X_i)$ be sequence of real independent r.v.'s and having the same law. If we let $X=\sum X_i$, how can one show that $(X_i, X)$ have the same law for any $i$. In this case $(X_i, X)$ is a random ...
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1answer
25 views

Correlation between two (correlated) normally distributed variables

In an electrical circuit I wish to find the noise correlation between two signal paths (which in the end are added). Let me also say, that I am quite useless at probability theory (but trying to ...
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1answer
23 views

Prove a simple property of conditional expectation

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F$, $\mathcal G$ and $\mathcal H$ be $\sigma$-algebras on $\Omega$ with $\mathcal F\subseteq\mathcal G\subseteq\mathcal H\...
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The existence of a moment for the product of an ergodic sequence of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of bounded stationary ergodic random variables with $$ \log(X_1) <0. $$ Then it can be easily shown, using Birkoff's ergodic theorem, that almost surely $$...
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Construct a stochastic process with independent stationary increments on dyadic numbers

I'm currently trying to construct a stoch. process with independent, stationary, normally distributed increments on the dyadic numbers. I guess the idea was given by Lévy. In order to do that I want ...
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17 views

Does conditional expectation exist if marginal expectation exists?

Given two random variables $X,Y$ on the same probability space, given that $E[Y]$ exists, does $E[Y|X=x]$ exist for all $x$ such that $f_X(x)>0$? My argument : no it does not. If $E[Y|X=x]$ is a ...
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1answer
27 views

Given a random variable $X_1$, does there always exsist an i.i.d. sequence $\{X_n\}$?

Below is from Tao's lecture note and he says there exists an i.i.d. sequence of random variables $\{X_n\}_n$ such that each $X_i$ is uniformly distributed. For any given random variable $X_1$, does ...
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0answers
34 views

Unbiased estimator of a square root of chi-squared distribution

Let $Y_1,Y_2,...,Y_n$ be a random sample from $N(\mu,\sigma^2)$. I need to show that $S$ is a biased estimator of $\sigma$. As from the definition, I see that $\frac{(n-1)S^2}{\sigma^2}\sim\chi^{2}_{(...
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1answer
30 views

Total waiting time of exponential distribution is less than the sum of each waiting time, how so?

I am reading my textbook and find a weird phenomenon. The example says that Anne and Betty enter a beauty parlor simultaneously. Anne to get a manicure and Betty to get a haircut. Suppose the time for ...
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1answer
18 views

Tail Probability limits

We see the series $\sum_{n=1}^\infty nP(X\ge n)$ show up a lot in probability theory, when dealing with expectations. I was wondering how to show that the terms tend to 0? ($nP(X\ge n)\rightarrow 0$)
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1answer
31 views

How do I prove that $E(f(X)) \leq E(f(Y))$ for $f$ increasing and $X$ and $Y$ multivariate Bernoulli?

Suppose $f:\{0,1\}^n \to R$ is an increasing function (we say that $x \leq y$ for $x,y \in \{0,1\}^n$ if $x_i \leq y_i$ for all $i$.) Let $X_p = (X_1, ..., X_n)$ be a random vector such that $X_i \...
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26 views

Brownian motion, first passage of time, strong markov property

So I need to find $P(max_{0\leq v \leq t} W(v) \geq 2$ and $min_{T_{2} \leq v \leq t} W(v) \leq -1)$. $T_{q}$ is the first passage time to level q My progress: If the first condition does not hold,...
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1answer
23 views

Ratio Distribution of Max and Min for Uniform random variables.

Let $X,Y$ be independent Uniform$(0,1)$ random variables. Set $Z = \frac{\min\{X,Y\}}{\max\{X,Y\}}$. Find the distribution of $Z$. So I need to calculate $\mathbb{P}(Z \le z)$. I wasn't sure if I ...
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1answer
19 views

Beta distribution CDF to Binomial Survival Function

There is a claim in my book that there is a connection to the Beta CDF and a Binomial Summation without explaining further. "Integration by Parts can be used to show that for $0<y<1$, and $\...
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1answer
17 views

Equivalence of conditions for Convergence of Non-negative Random Series

Question Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent. $\sum_{n=1}^\infty X_n<\infty$ a.s. $\sum_{n=1}^\infty[P(X_n>1)+E(X_nI(X_n\leq 1)]<\infty$ $...
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1answer
21 views

Independence of random variables and their components

If we have that $X$ and $Y$ are independent, what do we know about there components $X^\pm, Y^\pm$. Are they independent as well? If so, which combinations are the independent ones and which ones are ...
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1answer
18 views

Difficulty Understanding Solution to Marginal and Joint Density

I was able to solve part (a) using spherical coordinates. Part (b) is a completely different story. I'm not sure how the bounds were derived from the corresponding integral. Would someone on MSE be ...
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2answers
33 views

characterization of the existence of higher order expectation.

Consider a non-neg r.v $X$ and fix a positive integer $n$, I am trying to show that a necessary and sufficient condition for the $n$th order expectation to exist is that $\sum_{i\in\mathbb{N}}i^{n-1}P(...
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0answers
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Give an example of a continuous time stochastic process $X_t$ such that $E[X_t ] = 0$ but $X_t$ is not a martingale. [on hold]

Give an example of a continuous time stochastic process $X_t$ such that $E[X_t ] = 0$ but $X_t$ is not a martingale.
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Proving property of Markov chains.

There is a proof of the following Markov-chain I don't understand. I have circled in red the two steps I do not understand. Could you please explain the steps for me? In the proof they also refer to ...
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Proof of conditional independence properties

1) Definition of conditional independence: Let X1,...,XD be some random variables. For any A ⊂ {1,...,D}, let XA denote {Xa}a∈A, e.g. X{1,3} = {X1, X3}. Let A, B and C be three disjoint subsets of {...
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1answer
30 views

A random $r$-regular graph can be generated by taking union of a a random $(r-1)$-regular graph and a perfect matching.

$\newcommand{\lrp}[1]{\left(#1\right)}$ $\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\N}{\mathbb N}$ Definition. Let $(\Omega_n, \mc F_n)$ be ...
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0answers
18 views

Additivity of Independent variables

Given four variables $x, y, z, w$ such that $x$ is independent of $y$, and $z$ is independent of $w$. In general, do we have $x+z$ is independent of $y+w$? My instructor says it is true in class ...
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2answers
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Joint Density Formulas for XY and X/Y

I used convolution to find a formula for $X+Y$, but am unsure on how to figure out formulas for $XY$ and $X/Y$. Any help would be appreciated.
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Show inequality $X(t)\ge 3$ hold iff certain conditions hold.

Assume $\xi_i,\eta_j\sim\exp(1),$ and $$S=(-\eta_1,\xi_1,\xi_1+\xi_2,...,\xi_1+...+\xi_8),$$ that is a set with $9$ elements $s_1,...,s_9.$ Given $$f(t)=\left\{ \begin{array}{rcr} ...
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45 views

Why CDF is not coming to 1? [duplicate]

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
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which number will most comes out and which number will less? [on hold]

A box contains 1 to 10 digits .Pick one random choose in the box and replaced the number into the box . which number will most comes out and which number will less ? And it's solution depends on ...
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1answer
20 views

A question about a.s. convergence

While reading on almost sure convergence, I came across this equivalence: $$\{\omega \in \Omega: \lim_n X_n(\omega) = X(\omega)\} \equiv \bigcap\limits_{k=1}^{\infty}\bigcup\limits_{N=1}^{\infty} \...
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2answers
62 views

Expectation and linearity

Let $X$ be a non-neg random variable and define $A_i:=\{i-1\le X < i\}$ for each $i$. I have proved that $$\sum_i(i-1) I_{A_i}\le X <\sum_i iI_{A_i}$$ holds, but I have issues showing that it ...
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1answer
25 views

Density of absolute continuous measure wrt Lebesgue

I read the following theorem in my lecture material stating as follows: For any probability measure $\mu$ on $(\mathbf{R},\,\mathcal{B})$ with a density function $f(x)$, the following condition holds ...
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0answers
60 views

Show that $\int_{-\infty}^{\infty} f(t+x) \ dW(x) =\lim_{n\rightarrow \infty}\sum_{k=-x(n)}^{x(n)}f(t+\frac{k}{n})(W(\frac{k+1}{n})-W(\frac{k}{n})). $

I had this question up a few hours ago but it had lots of views but no answers, so I figured my question was bad and I did not provide enough information, so here is a better question I hope. In my ...
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1answer
18 views

Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$

Suppose $X$ is symmetric and stable with index $\alpha$ and $Y$ is stable and nonnegative with index $\beta$. Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$. The text also gives the ...
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0answers
13 views

Set of bounded elementary (simple) processes is a vector space

I am currently learning about stochastic integration with respect to semimartingales, and I am having trouble understanding every detail of the constructions, since I only learned stochastic ...
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1answer
19 views

Finding $a$ and $b$ such that $\hat u$ is an unbiased estimator

A chemist wants to decide the amount of a certain substance $\mu$ in a specific type of food. In the lab, the chemist has two measuring intruments $A$ and $B$. The results from the instruments can be ...