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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69 views

Show that X has density $f(x)$.

Let $$f(x)=60x^3(1-x)^2,\quad 0<x<1.$$ Let $U_1,U_2,...$ and $V_1,V_2,...$ i.i.d. random variables with distribution $U(0,1)$. We build a random variable X as follows: ...
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50 views

Find the almost sure limit

Let $X_1,X_2,...$ independent identically distributed (i.i.d.) random variables with $E(X_i)=2$ and $Var(X_i)=1$. Find the almost sure limit of: ...
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1answer
74 views

How to show that $X_n\to0$ in probability

Let $X_1,X_2,...$ independent random variables, with $P(X_n=n)=1/{n^2}$ and $P(X_n=1/n)=1-1/{n^2}$ . Show that $X_n\to0$ in probability. I got this problem in an old probability test but I'm not ...
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167 views

Limsup random variables not in $L^1$

Let $(X_n)$ be a sequence of real, independent identically distributed random variables in a probability space with probability function $\mathbb{P}$ such that $X_1 \notin L_1$. Prove that almost ...
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63 views

How can obtain the pdf of y?

I have a problem as follows: $$Y_k=N_k+AS_K \quad ,k=1,\ldots,n.$$$$$$where $\underline{N} \sim N(\underline{0},I)$ and where $S_1,\ldots,S_n$ are i.i.d. random variables, independent of ...
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246 views

Girsanov Transformation Example

Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion? For example computing: $E[(B_{n}+2n)^{2}]$ Set: $\widetilde{B_{n}}=B_{n}+2n$ And ...
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95 views

Fair coin toss probability asymptotics [duplicate]

Possible Duplicate: Asymptotics for a partial sum of binomial coefficients A fair coin is tossed $n$ times, let $A_n$ be the number of heads and $B_n$ the number of tails and $C_n = A_n - ...
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2answers
386 views

Sum of independent random variables almost surely constant

I am trying to solve the following problem: Let $(\Omega, \mathbb{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \ldots, X_n$ independent real random variables. Prove that the sum $X_1 + X_2 ...
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0answers
821 views

Hellinger distance between Gaussians - multivariate and univariate forms

On pages 46 and 51 of the book Statistical Inference based on divergence measures By Leandro Pardo Llorente there is a derivation for the Hellinger distance between two multivariate Gaussian ...
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1answer
87 views

Uniform integrability to use DCT

Suppose I have a family $F:=\{f_\alpha\}$, $\alpha \in J$ (index set) of positive functions, a function $L$ increasing, with values in $\mathbb{R}$ such that $L^+(F):=\{L^+(f_\alpha);\alpha \in J\}$ ...
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1answer
100 views

The Lebesgue integral $\int_\Omega dP$

I am a beginner. Given probability measure $P$ and sample space $\Omega$, is it true that: $$\displaystyle \ \ \int_\Omega dP = 1$$
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1answer
90 views

Probability that a sequence of random variables converges to 0 or 1

Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and $$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) ...
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1answer
86 views

What is the name of this theorem, and are there any caveats?

For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ ...
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1answer
88 views

Are there distributions that satisfy this equation for more functions?

For random variable X following any distribution, the following is true for any function $\phi(x)=ax+b$, where $a$ and $b$ are constants: $$\mathbb{E}[\phi(X)] = \phi(\mathbb{E}[X])$$ Are there ...
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1answer
155 views

Question about of Fatou's lemma in Rick Durrett's book.

In Probability Theory and Examples, Theorem $1.5.4$, Fatou's Lemma, says If $f_n \ge 0$ then $$\liminf_{n \to \infty} \int f_n d\mu \ge \int \left(\liminf_{n \to \infty} f_n \right) d\mu. $$ ...
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271 views

how can I get minimum error probability for this decision problem?

I have the decision problem for 4 hypotheses as follows: $$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$ where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$ In ...
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1answer
32 views

general coupling of a projection

If $c$ is a coupling between two measures, $c= \mu^1\, \, t \, \, \mu^2$, ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both defined ...
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44 views

distance between measures, proof of an inequailty

Let $\Omega = \prod _{s \in S} \Omega_s$, with $\Omega_s$ finite and all the same, $S$ countable. Let $\mu_1$ and $\mu_2$ be two probability measures on the product space (not necessarily the product ...
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1answer
249 views

Finite almost surely implies integrable?

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. If a random variable $X$ satisfies $\mathbb{P}[X<\infty]=1$ (which means $X$ is finite almost surely, doesn't it?) then ...
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2answers
89 views

Asymptotics for $\mathbb{E} [|X_1|^r . I_{|X_n| > b_n}]$

Let $X_1, X_2, ...$ be a sequence of i.i.d. random variables such that $\mathbb{E}|X_1|^r < \infty$ for some $r > 1$. If b_n is a sequence such that $b_n \to \infty$ as $n \to \infty$ then ...
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1answer
199 views

How Borel sets make $\sigma$-algebra in a topological space?

I am trying to wrap my head around random variables and can't prove the following questions: How, in a topological space $(X, \mathcal{T})$, the collection of all Borel sets, say $\mathfrak{B}$, ...
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1answer
59 views

coupling of the projection and projection of the coupling

Let $S$ be a countable space, $\Omega_s$ be a finite space and $\Omega_S = \prod_{s \in S} \Omega_s$ be the product space, equiped with the product topology. Let $\mu^1$ and $\mu^2$ be two probability ...
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1answer
117 views

Bounded variation and continuous local part when using Ito's Formula

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...
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2answers
116 views

Proving an asymptotic property regard the fraction of ‘1’ and ‘0’ in binary sequences

Consider the set of sequences of zeroes and ones of length $N$ with $k$ ones (or, $Np$ ones where $p = k/N$). We draw randomly and uniformly a sequence from this set. I want to show that with ...
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1answer
47 views

About Conditional Independent

I know that independent and conditional independent don't imply each other. But what if given more condition that $Z$ is independent from $X$ and $Z$ is independent from $Y$? So the problem is: A: ...
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2answers
123 views

Finding a High Bound on Probability of Random Set

first time user here. English not my native language so I apologize in advance. Taking a final in a few weeks for a graph theory class and one of the sample problems is exactly the same as the ...
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2answers
141 views

How do wer find this expected value?

I'm just a little confused on this. I'm pretty sure that I need to use indicators for this but I'm not sure how I find the probability. The question goes like this: A company puts five different ...
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1answer
114 views

Where are algebras and $\sigma$ algebras studied?

I am taking a course on real analysis (mainly about Lebesgue measure etc') and a few lectures back the lecture introduced to concept of algebra and $\sigma$-algebra. It feels a bit strange to see it ...
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1answer
400 views

Convergence in Central Limit Thorem

The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not ...
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57 views

measurability of a random variable coming from a.s. equivalence

Suppose you have a random variable $X_1:\Omega \to \mathbb{R}$, which is $\mathcal{F}$ measurable and a random variable $X_2:\Omega \to \mathbb{R}$, which is $\mathcal{F}_t$ measurable with ...
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78 views

What is the expected number of point in time it is in room $2$

A man is forced at time $0$ into a five-room maze shown as the diagram. At the end of each unit of time, it changes to a different room by choosing an exit at random. Let $X_n$ be the room number ...
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421 views

A question about how to get the limiting probability.

Suppose $p=\begin{bmatrix} 0& 1\over 3 &0 &2\over 3 \\ 0.3& 0& 0.7 &0 \\ 0& 2\over 3&0 &1\over3 \\ 0.8& 0& 0.2& 0 \end{bmatrix}$is the ...
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123 views

Finding the first and second moments from mgf and finding general formula for nth moment

Suppose that the random variable Y has the moment generating function: $M_y(t) = \frac{1}{5}(e^{-2t} + e^{-t} + 1 + e^t + e^{2t})$ for all real t. 1) Find $E[Y]$ and $VAR[Y]$. 2) Give a general ...
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92 views

Exercise of weak convergence

This is an exercise of Central Limit Theorem (CLT): Let $(X_j)_{j\geq 1}$ be i.i.d. with $E[X_1]=1$ and $\sigma_{X_1}^2=\sigma^2\in(0,\infty)$($\sigma>0$). Show that $$ ...
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262 views

Is this problem just really long or am I doing something wrong?

I am to show that $$f_{XY}(x,y)= \begin{cases} c(y^2-x^2)e^{-y}, & (-y \le x \le y),\quad y\gt0 \\ 0, & \text{otherwise} \end{cases}$$ has gamma density and show that $c=\frac18$. I know I ...
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2answers
63 views

How do we represent this event?

My professor did this problem out of our text book and he didn't exactly show us how he did it (he skipped showing the steps in the middle). He got the answer $(1-x)^2$. If we let $X$ and $Y$ be ...
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185 views

Does Kolmogorov 0-1 law apply to every translation invariant event?

Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ ...
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1answer
528 views

Expected value of Max of correlated random variables inequality

I have $\xi$ and $\eta$ with following properties: $E\xi = E\eta = 0$, $D\xi = D\eta = 1$. And the correlation coefficient: $\rho = \rho (\xi, \eta)$. I want to prove the following inequality: $$ E ...
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1answer
246 views

Trying to find P(X=k|Z=m) for binomial random variables with parameters n and p

Let $X$ and $Y$ are independent binomial random variables with parameter $n$ and $p$. Their sum is denoted by $Z$. How do I prove : $$ P(X=k\mid Z=m) = {n \choose k}{n \choose m-k}\biggm/ {2n \choose ...
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1answer
516 views

Conditional Markov inequality

Is there a conditional Markov inequality? I mean, assume that $X$ is a random variable on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$, and $X\geqslant 0$. Is then ...
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1answer
325 views

Exponential Probability Monte Carlo simulation

I need to write a Matlab program to estimate the quantity $\theta = \mathrm{Pr}(X < 1)$, where $X$ is an exponential random variable with mean $1$. I am doing this for multiple monte carlo ...
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83 views

Does the absolute value even matter here?

In this problem I'm doing it says Suppose that $(X,Y)$ is uniformly distributed over the region {$(x,y):0\lt |y|\lt x\lt 1$}. Find the marginal densities $f_X(x)$ and $f_Y(y)$ Does the absolute ...
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1answer
2k views

CDF of sum of dependent random variables

Suppose that $X$ and $Y$ are $dependent$ random variables, what would be the cumulative distribution of $X+Y$? That is, what is $P(X+Y\le c)$ for any integer c? Note that we do not know their joint ...
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105 views

Examples that are not Lebesgue integrable for any $p$

I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
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1answer
33 views

Finding conditional probability

Given $P(A)$, $P(B)$, and $P(B\mid A^c)$, how do you find $P(B\mid A)$? I need this to find $P(A\mid B)$ using Bayes' Theorem: $$ P(A\mid B)=\frac{P(A)P(B\mid A)}{P(A)P(B\mid A)+P(A^c)P(B\mid A^c)} ...
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152 views

How do I find a formula for this?

Yes this is homework and no I'm not looking for direct answers, just tips or some clairification on how to do this problem since it doesn't make sense to me and my professor didn't explain it very ...
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1answer
52 views

Conditioning on a Measurable and non-Measurable Space

$\mathcal{F}$ and $\mathcal{G}$ are two sigma algebras on the same space, neither a subset of the other. If $X$ is $\mathcal{F}$-measurable but not $\mathcal{G}$-measurable is: $$ E[E[X\mid ...
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255 views

Why do we require absolute convergence in the definition of expectation?

I am studying some basic probability and the lecture have defined the expectation of a $R.V$ ,$X$, in the cases that $X$ is discrete or continues. In both definitions we first required some absolute ...
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3answers
63 views

Simple question: Why does $E(|X|) < \infty$ imply $E(|X|I_{|X|>a} )$ tends to $0$ as $a$ tends to infinity

Simple question: Why does $E(|X|) < \infty$ imply $E(|X|I_{|X|>a} )$ tends to $0$ as $a$ tends to infinity? I've seen it in a few proofs and I can't see why this is the case, I've tried a proof ...
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1answer
42 views

A simple question about probability

Assume that on average, only $1$ in $80$ calls made by a teleseller can he/she approaches a potential client. So 1.) what is the probability that a teleseller fails to approach any potential client ...