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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23 views

Can anyone help me find the variance of this expression?

I have a vector of the form \begin{align} {\bf a }= \frac{1}{\sqrt{N}}[1, e^{jA}, e^{j2A},\cdots, e^{j(N-1)A}]^T \end{align} where A and N are constants. I also have a vector N of i.i.d ...
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31 views

Convergence of unbounded functionals

Let $\mu_n$ be a sequence of random measures on the metric space $(\mathbb{R},d)$ that converges in some mode to a measure $\mu$. The definition of weak convergence is: \begin{equation} ...
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3answers
60 views

Why is a probability density function nonnegative?

Let $X$ be a random variable and its density $f$ be defined to be the derivative of its distribution function $F$, i.e. $$\Pr(a< X\le b)=F(b)-F(a)=\int_a^bf(x)\operatorname{dx}$$ Now let ...
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1answer
39 views

Expectation of truncated Poisson Distribution

I have found that if I have a $Y \sim \mathrm{Poi}(\lambda)$ and $Z=Y \mid Y>0$ then I say $$f_Z(k)=g(k)=Pr(Y=k\mid k>0)=\frac{\lambda^k}{k! (e^\lambda-1)}$$ Now I am trying to compute ...
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1answer
13 views

How to find the mean variable of a normal distribution with a given probability and standard deviation?

We have a machine that produces µ g of pasta to be stored in their package, with a standard deviation of 20g. It follows a normal distribution. And we don't want it to produce more than the package's ...
2
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1answer
20 views

A simple question about the definition of martingales

The definition of Martingale denotes that $E(M_{n+1}|\mathcal{F}_n)=M_n$. This implies $E(E(M_{n+1}|\mathcal{F}_n))=E(M_n)$. Then does it mean that $E(M_{n+1})=E(M_n)$ using the tower property? If so, ...
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10 views

Application of Strong Markov Property

Theorem SMP (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or $\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many values. Then ...
2
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1answer
35 views

Skorokhod's theorem and summary of convergence of sequence of RVs

This question is about convergence of RVs (when convergence in one sense implies other convergence modes). I would like to have a big picture on convergence modes and various implications between ...
5
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1answer
48 views

Prove that this sequence converges almost surely

Suppose that $(X_n)_{n\ge1}$ is a sequence of independent random variables with $E[|X_n|] < \infty$ for all $n$ and $E(X_n) = \mu$. Prove that $$\sum_{n=1}^{\infty}\frac{1}{2^n}X_n = \mu \; a.s$$ ...
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2answers
20 views

7 white balls and 3 black balls in a box. You take out in sequence 3 balls at random without replacement. Probability that all 3 balls are black?

What i did was say that the probability the first ball is black is 3/10, then since we removed one the probability the second is black is 2/9 and third is ⅛. So then the probability all three are ...
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2answers
16 views

Marginal P.M.F and Conditional Expectation?

I have a joint density function that is formulated as follows: $$ f_{X,Y}(k,y) = \begin{cases} \frac{\partial{P(X=k, Y\le y)}}{\partial y} = \lambda \frac{(\lambda y)^k}{k!}e^{-2\lambda y} & ...
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0answers
48 views

Concept of uniqueness of almost sure convergence

I want to show the following statement: If $X_n \rightarrow Y_1$ in probability and $X_n \rightarrow Y_2$ in probability also. Then $Y_1 = Y_2$ almost surely. Here is my approach: I want to ...
2
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1answer
29 views

Subset and optional times

The below is a well known fact but can anyone help me to prove it? If I have a right continuous filtration and $\eta$ is an optional time, how can I show that if $\eta\leq t$ then $\mathcal{F}_\eta ...
2
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1answer
60 views

Showing that $(1-u)z^2\leq P(uz\leq |X|)$ when $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$.

I am trying to show that $$(1-u)z^2\leq P(uz\leq |X|)$$ where $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$. I've been given a hint to consider Cauchy-Schwarz, however, I don't see where ...
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0answers
23 views

Convergence in probability (in measure) of a weighted sum of random functions

Consider a mesh of points $\pi_{n} = (t_{1n},\ldots,t_{K_{n}{n}})$ with $0 < t_{1n} < \ldots < t_{K_{n}n} < 1$ and weights $(w_{1n},\ldots,w_{K_{n}n})$ such that ...
2
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0answers
20 views

Stratified Sampling: Total and mean estimate of population

Question: In a sample survey designed to estimate the total number of cattle, the universe of 2072 farms was stratified into 5 strata on the basis of the total acreage of farms. In the hth stratum (h ...
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3 views

Power spectral density and measure in continuous time

Let $X_t$ be a continuous-time wide-sense-stationary stochastic process. The name of the power spectral density (defined as the Fourier transform of the auto-correlation function) suggests it can be ...
4
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1answer
46 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
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1answer
16 views

How can we derive cross covariance $R_\mathrm{xy}(t_1,t_2)=R_\mathrm{yx}^*(t_2,t_1)$?

In random process, cross covariance is nonnegative definite like $$R_\mathrm{xy}(t_1,t_2)=\mathbf{E}(\mathrm{X}(t_1)\mathrm{Y}^*(t_2))=R_\mathrm{yx}^*(t_2,t_1)$$ I'm wondering how it can be derived. ...
2
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1answer
81 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
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1answer
81 views

Prove that the maximum of $n$ independent standard normal random variables, is assyptotically equivalent to $\sqrt(2\log n)$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log ...
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1answer
34 views

Why are linear combinations of independent standard normal random variables also normally distributed?

My professor has given a list of questions that will not be appearing on my test, with this being one of them. I still feel this is extremely important to understand. How can I prove the following ...
0
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1answer
28 views

Probability and Measure: Sigma-finite

What is a example that shows that $\mu$ $\sigma$ -finite does not imply $\mu \cdot T^{-1}$. I have a basic understanding of what $\sigma$-finite means but if $\mu$ is finite implies $\mu \cdot T^{-1}$ ...
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0answers
39 views

Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory

I'm taking stochastic probability class but I'm now only taking analysis (with Rudin's PMA) class. The stochastic probability class doesn't depend heavily on the theoretic structures: rather, the ...
0
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1answer
34 views

Coin toss probability

Three coins but into hat. Coin 1 shows heads on both sides, coin 2 is a fair coin, and coin 3 has a .75 probability of landing heads...one coin is randomly selected and flipped. The coin lands heads ...
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0answers
53 views

arranging people into seats in a row

Suppose there are 100 people in a line and they are to be arranged into 100 chairs in a row. Each of them has already selected one number $x_i$ from $1$ to $100$ randomly (i.e. all numbers with equal ...
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2answers
41 views

I need to find the smallest lambda for which $P(X\ge 2)=0.99$ when $X\sim \text{Poisson}(λ)$

What I have tried to do is this: Since $P(X>=2)=0.99$, then $P(X<2)=0.01$, Hence $P(X=0)+P(X=1)=0.01$, so replacing in the pmf, I got $\exp(-λ)+λ\exp(-λ)=0.01$, $g(λ)=\exp(-λ)+λ\exp(-λ)-0.01$. ...
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21 views

pairwise and mutual independence

I need to show that that if $A_1, A_2, A_3$ are independent then $A_1, A_2, A^c_3$ are independent, where $A^c_3$ is the complement of $A_3.$ I am bad at proofs and would love some help with this ...
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18 views

Devising probability triple for coin toss experiment

We are flipping a coin until we get a head. The task is to devise a probability triple. So I am thinking the sample space could be $$ \Omega = \mathbb N $$ where each outcome denotes the no. of flips ...
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2answers
32 views

Independent integrable random variables with 0 expectation so that $\overline{S}_n$ does not converge to 0 in probability

Give an example of independent integrable random variables $X_n$ such that $E[X_n] = 0$ for all n, but $\overline{S}_n = (\sum_{i=0}^n X_i)/n$ does not converge to 0 in probability. As far as I ...
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1answer
33 views

The quantile function $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$ has the condition that $Pr(X=c) = p_1 - p_0$

Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions: (i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$, (ii) ...
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2answers
39 views

Cauchy convergence in probability implies the existence of a (finite a.e.) limit X

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an X (finite a.e.), such that $X_n$ converges to X in probability. The problem's hint suggests ...
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0answers
25 views

Covariance of states of a finite Markov chain

I know it is possible to construct a covariance matrix for states of a Markov chain but I cannot seem to find a proper way to compute it. I will attach some theories I found from Kemeny and Snell's ...
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2answers
49 views

What's the different among the concepts Probability, Possibility and Belief? [closed]

Can you please explain the difference among the concepts Probability, Possibility and Belief through some simple examples?
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3answers
51 views

Finding $P(N>E(N))$

I know how to do the (i) and I will put out the results, just in case the second part is related to the first part answers. The C.D.F : $$F(t) = 1 – (t – 1)^{-2}$$ $P(T>5)$ : $$= 1 – F(5) = 1 ...
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0answers
31 views

$X^{T}AX$ and $BX$ are independent when $BA=0$

Suppose $X \sim \mathcal{N}_k(0,I_k)$. Let $A$ be a $k \times k$ symmetric idempotent matrix and $B$ be a $l\times k$ matrix of full row rank. Show that if $BA=0$, then $X^{T}AX$ and $BX$ are ...
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0answers
24 views

A generalization of SLLN

Let $\{X_i\}_{i=1}^\infty$ be a sequence of iid integrable random variables. Let $A \in \mathbf{B(\mathbb{R})}$. Does the following result hold (with possibly some more assumptions like $P(X_1 \in A) ...
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2answers
33 views

Condition on two distributions. All N(0,1).

This problem The RV $X_1$,$X_2$,$X_3$ are independent $N(0,1)$. Consider $Y_1=X_2+X_3, Y_2=X_1+X_3,Y_3=X_1+X_2$. Find the conditional density of $Y_1$ given $Y_2=Y_3=0$ (From Gut An intermediate ...
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1answer
33 views

Condinational Probability of a multinomial random variable

Let $\mathbf X=(X_1,X_2,X_3)$ be a multinomial random variable having the probability density function defined by: $$f_{\bf X}({\bf x})=\dfrac{n!}{x_1!x_2!x_3!}p_1^{x_1}p_2^{x_2}p_3^{x_3}$$ with ...
2
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1answer
50 views

probability of a limiting sum

Suppose that $U_i$ are uniformly distributed on (0,1) and are independent. For all possible increasing index sets comprising the family $J$, I am trying to show that $P(\cap_{j\in J} \{\lim_{n ...
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0answers
37 views

Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
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1answer
50 views

Prove $\operatorname{Var}(\hat{e}_{ij}) = \sigma^2 \left(\frac{n_i-1}{n_i}\right)$

$\newcommand{\Var}{\operatorname{Var}}$ Let $y_{ij}$ denote the observed response of the $j$th experimental unit in the $i$th treatment group, and the $e_{ij}$ are i.i.d. $N(0,\sigma^2)$ experimental ...
0
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1answer
65 views

PDF and CDF of probability theory [closed]

The continuous random variable X has pdf $$f(x) =\begin{cases} x/2, \ 0<=x<=2 \\ 0, \ \text{elsewhere} \end{cases} $$ Two independent determinations of X are made. What is the probability ...
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1answer
15 views

An inequality in a step in the proof of monotone convergence for random variables

I'm proving monotone convergence for random variables, i.e. If random variables $X_n$ are positive and increasing almost surely to $X$, then $$\lim_{n\rightarrow ...
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1answer
33 views

Conditional expectation of the maximum of two independent uniform random variables given one of them

Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$. What is the conditional expectation of $\max\{X_1,X_2\}$ given $X_2$? And the conditional expectation of ...
0
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1answer
25 views

Expectation of a random functional

Suppose that $X_{1},\ldots,X_{n}$ are i.i.d. random variables and let $\hat{F}_{n}$ be a random cdf that depends on $(X_{1},\ldots,X_{n})$. For instance $\hat{F}_{n}$ could be the empirical ...
2
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1answer
38 views

Weak version Fatou lemma

I want to show the weak version Fatou Lemma; i.e., Let $f \ge 0$ be continuous function. If $X_n \rightarrow X$ in distribution, then $$ \liminf_{n\rightarrow \infty} E f(X_n) \ge Ef(X) $$ ...
1
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1answer
35 views

$\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$ implies that random variables $X_n$ are uniformly integrable

$X_n$ are uniformly integrable if $\lim_{R \rightarrow \infty} \sup_n E[|X_n|,|X_n| \geq R] = 0$. Show that if $\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$, then $X_n$ are uniformly integrable. ...
0
votes
1answer
24 views

Expected Value of the value of the width of the interval $(X,4X)$

Find the Expected Value of the value of the width of the interval $(X,4X)$ How do I find the Expected Value in this case since the interval consists of X, the problem is that the answer is an ...
0
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0answers
26 views

Almost sure downward convergence with some conditions implies convergence in $L^1$

If $X_n\downarrow X$ a.s., each $X_n$ is integrable and $inf_n E[X_n] > -\infty$, then $X_n \rightarrow X$ in $L^1$. As far as I know, "$X_n\downarrow X$ a.s." means that for every n, $X_n ...