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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123 views
Combinatorics challenge [closed]
A person stands on an imaginary circle with a radius R facing the center of the circle O. The person can make a step to the left or to the right and then again step on the circle. The average length ...
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0answers
30 views
Why is K-L divergence defined as it is?
Why is the K-L divergence defined this way:
if $P$ and $Q$ are probability measures over a set $X$, and $P$ is absolutely continuous with respect to $Q$, then the Kullback–Leibler divergence from ...
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1answer
58 views
Lower Expectation
Let $X$ be, for simplicity, a finite set (with the discrete topology).
Denote with $M(X)$ the set of probability measures on $X$ endowed with the weak topology.
For $\mu\in M(X)$ and a (necessarily ...
1
vote
1answer
43 views
Equality of two limits of r.v.
considering a sequence of real-valued r.v. $(X_n)$ convergent to $X$ in probability. Moreover we look at a sqeuence of r.v. $(Y_n)$, where $Y_n\in\operatorname{conv}(X_n,X_{n+1},\dots)$ and we suppose ...
2
votes
1answer
33 views
Martingale equality
The question is to prove
$$P\{\sup_{t\geq 0}M_{t}>x\mid \mathcal{F}_{0}\}=\min\left\{1,\frac{M_{0}}{x}\right\},$$
where $M$ is a positive continuous martingale which converges to 0 almost surely ...
1
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0answers
15 views
Analytic random function
Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function.
I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic.
What are the minimal conditions needed? ...
7
votes
2answers
55 views
Two martingales whose distributions agree for each time have the same overall distribution
Let $\{X_n\}$ and $\{Y_n\}$ be two martingales. Suppose that for each fixed $n \in \mathbb Z_+$, $X_n$ and $Y_n$ have the same distribution. Must it hold that the random sequences $\{X_n\}$ and ...
1
vote
0answers
23 views
What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]
Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads?
How do I apply Chernoff Bounds to this?
I really need help understanding Chernoff Bounds.
0
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1answer
26 views
Question regarding Type II Error in Hypothesis Testing
The following is a homework problem and I am not really sure where to begin or how find what the question is asking.
Suppose that one observation from the exponential pdf $f_{y}(y)=\lambda ...
3
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0answers
58 views
Relation between factor graph and conditional probability distribution
First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me.
The question
Let say I have a factor graph illustrated in the figure.
The ...
2
votes
1answer
46 views
Using empirical density function as an estimator of a given probability density
We know empirical distribution function is defined as $F_n(x)=\frac{1}{n}\sum\limits_{i=1}^nI(X_i \leq x)$. Then define empirical density function as $ f_n(x) = \frac{F_n(x+b_n)-F_n(x-b_n)}{2b_n} $ .
...
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1answer
70 views
How to sample uniformly from an $\epsilon$ ball?
Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed ...
0
votes
1answer
44 views
Convergence to a constant in probability but not almost surely
Please give a example that a sequence of random variables that converge to a
constant $c$ in probability but fail to converge to $c$ with probability $1$.
Thanks very much.
1
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0answers
35 views
Expectation of Random Variables - Measure Theory
I am trying to do the exercise 2 of section 3.2 of the book "A Course in Probability Theory by Kai Lai Chung". Problem asks to show:
If $\mathscr E(\left|X\right|)<\infty$ and $\lim_{n\to ...
3
votes
1answer
38 views
Bayesian formula for weather exercise
If it is nice weather on one day, the probability that it is going to be nice again the next is $13/15$.
If it is raining on one day, the prob. that it is going to be raining again the next day is ...
3
votes
1answer
38 views
poisson distribution jobs in printer
A printer receives a number of jobs in an hour, which is poisson distributed with parameter $\lambda$. Every job is recognized with a probability $p$ such that the job is faulty and wont be printed.
...
1
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1answer
32 views
Bernoulli trials conditional probability
Let $\Omega=\{0,1\}^\infty$ and $S_n=X_1+\cdots+X_n$ the number of “successes” or “arrivals” in $n$ steps. $p\in(0,1)$ and $\mathbb P(S_n=k)=\binom{n}{k}p^k(1-p)^{n-k}$
Let $T$ be the time until the ...
0
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1answer
23 views
Generalize the invertible stochastic matrices (Markov chain) as a group.
Define $\sum'(2, \mathbb R) = \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_n (\mathbb R) : a + c = 1,\ b + d = 1 \}$.
Then clearly $\sum'(2, \mathbb R)$ with the matrix ...
1
vote
0answers
57 views
Probability distribution with maximal entropy on $[0,1] \cup \{2\}$
For given closed set $F$ on $\mathbb R$ one can think of probability distribution $\mathbb P^\ast_F$ with support on $F$ and with maximal entropy. It is well known that
If $F=[0,1]$ then $\mathbb ...
1
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1answer
71 views
Martingale Stopping Time
Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | ...
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0answers
27 views
When a family of measures provide continuity?
Consider a mapping $m: \mathcal{B}(X) \times P \rightarrow [0,1]$, where $X \subseteq \mathbb{R}^n$, $P \subseteq \mathbb{R}^m$, and $\mathcal{B}(X)$ denotes the Borel sets.
$\forall p \in P$, ...
1
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0answers
40 views
asymptotic order of the variance of the maximum of iid standard Gaussian
Suppose that $X_1,\cdots,X_n$ are iid standard Gaussian. $X_{(n)}$ is the maximum of $(X_1,\cdots,X_n)$, how can I find the asymptotic order of $VAR[X_{(n)}]$?
The density function of $X_{(n)}$ can ...
0
votes
1answer
35 views
Finding that probability of the event is small
Let $x_1, \ldots, x_n$ be Bernoulli random variables with the probability of success $P(x_i=1)=p$.
Let $\epsilon>0$.
Show that probability
$$
P\left(\left|\sum_{i=1}^nx_i-p\right|> ...
1
vote
1answer
52 views
Two definitions of Bayes Sufficiency
"Bayes Sufficiency" is defined in two ways. Are they equivalent?
Setting
A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
1
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1answer
30 views
Hypothesis Testing a small sample for the binomial parameter p
The following is a question from a homework set that I truly do not understand how to even begin.
The following is a Minitab printout of the binomial pdf $p_{x}(k) {9 \choose k}(0.6)^k(0.4)^{9-k}$, ...
3
votes
1answer
69 views
Random Walk on Z
Let $S_n$ be the symmetric random walk on $\mathbb{Z}$. How do i calculate
$P(\limsup_{n\rightarrow\infty} S_n=\infty)$?
I already know that the probability is 1 but I don't really know how to start? ...
0
votes
0answers
19 views
Power Curves from Normal Distribution.
The following is a homework problem that I cannot figure out because I am having trouble finding the Type II error.
Construct a power curve for the $\alpha = 0.05$ test of $H_0:\mu = 60$ versus $H_1: ...
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0answers
38 views
Density function $e^{{-x-e}^{-x}}$ [duplicate]
Let $f(x) = e^{{-x -e}^{-x}} $ .
How can I check that $f$ is a density function?
I know that it has to be valid that $ \int_{-\infty}^{\infty}{f(x)} = 1 $ , but how to check this?
Thanks a lot
2
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1answer
53 views
Why every strict stationary process have the following representation
Suppose $\{X_n\}_{n\geq 1}$ is a strict stationary process, that is for every $n$, $(X_0,\ldots,X_n)$ and $(X_1,\ldots,X_{n+1})$ have the same distribution.
Then there is a probability space ...
2
votes
1answer
60 views
Smallest $\sigma$-algebra generated by $\mathcal C$?
Can someone explain how this $\sigma$-algebra is attained? It's mainly the $X\cup Y$ bit which I don't understand.
Question: If
$\Omega = \{1, 2, 3, 4\}$ and we have a collection of sets
$\mathcal C ...
2
votes
1answer
60 views
$\sigma$-algebras and independent stochastic processes
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
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1answer
25 views
Probability 4 colored bags & 4 colored balls
I came across this problem, and I do not understand how to solve it. Do I have to make an exhaustive list, or is there a simpler method?
QN:
There are 4 differently colored balls (red, blue, ...
1
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1answer
54 views
Transforming a submartingale into a supermartingale
Consider the following model.
$X_{n+1}$ given $X_n, X_{n-1},...,X_0$ has a Poisson distribution with mean $\lambda=a+bX_n$ where $a>0,b\geq{0}$. If $b\geq 1$, then $E[X_{n+1}|X_n]= a+bX_n > ...
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0answers
45 views
A problem in the coupling of Markov chains
Let $\{X_n\}$ and $\{Y_n\}$ be two independent discrete Markov chains with the same state space $S$ and the same transition probability $P$. $X_n$ has initial distribution $\mu$ and $Y_n$ has initial ...
6
votes
1answer
112 views
Conditional expectation on more than one sigma-algebra
I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
1
vote
1answer
36 views
Convergence of sum of random variables
Let $X_n$, $n\geq 0$, be i.i.d. random variables such that: $\mathbb E(X_1)=0$, and $0<\mathbb E(|X_1|^2)<\infty$. Given that $\alpha >\frac{1}{2}$, I need to show that ...
1
vote
1answer
45 views
maximum of exponentials
I am really having difficulties to prove the following: consider $X_1,\dots, X_n$ all exponentially distributed with rate $\lambda$ (i.e. $X_i \sim exp( 1/\lambda)$). Then argue that we can write
...
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votes
3answers
92 views
Double conditional probability
Is it possible to compute $P(X\mid Y,Z)$ by calculating $P(X\mid Y)$ given the probability $P(\cdot\mid Z)$? Similarly, is it possible to get at the density $f_{X\mid Y,Z}$ by calculating the desity ...
5
votes
2answers
78 views
Does there exist a finite fair gamble game with one dishonest coin?
I am thinking, maybe a well known problem, of
whether there exists a fair gamble game for two persons by tossing one dishonest coin that will always stop(one winner is selected) at no more than $N$ ...
1
vote
2answers
43 views
Computing a conditional expectation
I'm trying to compute a conditional expectation. If $(\Omega, \mathcal{F}, P)$ denotes a probability space, and let $A, B\in\mathcal{F}$ with $0<P(B)<1$ and let $\mathcal{G}=\{ B, ...
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2answers
27 views
Positivity of Conditional Expectation
This might be a very easy to answer question: Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $\mathcal F'$ a $\sigma$-subalgebra of $\mathcal F$. Let $X:\Omega\rightarrow\mathbb R$ be ...
0
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1answer
35 views
How to define a mixed strategy in a game with a countable action space
Suppose in a two player zero-sum game, player I chooses a number $m$ from $\mathbb{Z}$, and player II chooses $n$ from $\mathbb{Z}$.
If $m-n =1$, then player I recieves a payoff of $1$, while ...
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0answers
49 views
something about property of Bernoulli random variables
Let $b_i, i=1, \ldots, n$ be Bernoulli random variables with probability $P(b_i=1)=2k/n,$ where $k\leq n.$
Show the following:
Let $\chi$ be an indicator function that $k$ out of $n$ of $b_i$ are ...
0
votes
1answer
56 views
Conditional expectation $E[X|Y<y]$
Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.
Is it true that $E[X|Y < y]$ equals:
$$ \displaystyle ...
0
votes
1answer
40 views
Probability that the absolute value of one random variable is less than the absolute value of another
I'm having trouble wrapping my brain around this idea.
Suppose we have $X$ and $X'$ which are IID continuous random variables with conditional cdfs:
$F_{+}(x) = P(X \leq x | \epsilon = 1)$
...
2
votes
2answers
35 views
Finding sequences such that function of sum of r.v's is martingale
Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle S_{n}=\sum_{i=1}^{n}X_i$ and ...
2
votes
1answer
49 views
Nonnegative Superharmonic Function is Constant for $d>2$?
I have to do the following:
Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
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1answer
36 views
Need help with the following:
Proof or counterexample:
a) if $T$ is ergodic, then $T^2$ is ergodic, b) if $T$ is strong mixing, then $T^2$ is strong mixing.
Thank you.
3
votes
1answer
56 views
Convergence of $L^p$ norm as $p \downarrow 0$ [duplicate]
Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in ...
0
votes
2answers
50 views
Central Limit Theorem VS Normal Model
I just had a quick question regarding the Central Limit Theorem and Normal Model. I am in an elementary probability course and we have learnt that the CLT is as follows:
$$ Z = \frac{X_1 + X_2 + X_3 ...
