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 ...

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1answer
15 views

Poisson counting process question, but correct answer not obtained by usual method

Ok here's the question: Fisherman Dan is out fishing by a stream. On average, 3 fishes per hour swim by but Fisherman Dan catches the fish with probability 1/2. It rains in average once per ...
0
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1answer
48 views

Prove absolute sum expectation

I have encountered the following problem, could someone provide me some hints on how to solve it? Assume that the sequence $(X_n)$ is i.i.d. with mean $0$ and variance $1$. For every $n\ge1$, let ...
3
votes
1answer
20 views

Independence of random variables and covariance in the limit.

Consider two sequences of random variables $(X_n)$ and $(Y_n)$ which converge in distribution to $X$ and $Y$ respectively, where $X$ and $Y$ are independent, but each pair $(X_n, Y_n)$ is not ...
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2answers
31 views

Split 1 in to n parts of size $2^{-k}$

I have the following problem: Let $n \geq 2$. Let $p_{i} = 2^{-k_{i}}, k_{i} \in \mathbb{N}$ s.t. $$p_{1} \geq p_{2} \geq ... \geq p_{n}$$ and $$\sum_{i=1}^{n} p_{i} = 1$$ I have to show the ...
0
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0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
1
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0answers
24 views

One dimensional Lazy random walk, $O(1/\sqrt{n})$?

Suppose that we have a Lazy 1-dimensional random walk $X_n$ valued in $\mathbb{Z}$, i.e. $$X_n = \sum_{i}^{n} \xi_i\;\;\;\;\;\;\;\;(\xi_i\;\text{iid}) $$ and $$\frac{1}{4}=P(\xi_1= 1)=P(\xi_1 =-1) ...
1
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1answer
17 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
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1answer
51 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
2
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0answers
35 views
+100

Mean value theorem for random variables

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$f(X+Y)=f(X)+f^\prime(X+\theta Y)(Y-X)$$ for real valued random variables $X$ and $Y$ and ...
3
votes
0answers
16 views
+50

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
1
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2answers
23 views

Are as constant but not constant random variables trivial sigma-algebra-measurable? Converse?

Are almost surely constant random variables trivial sigma-algebra-measurable? These links suggest no: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2004&task=show_msg&msg=1121.0001 ...
3
votes
1answer
22 views

Kurtosis of sum of Independent Random Variables

Suppose that $X$ and $Y$ are independent random variables with different expected values and variances. Suppose we define kurtosis as $$Kurt(X)=\frac{E[(X- \mu)^4]}{E[(X- \mu)^2]^2}$$ My question is ...
0
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2answers
34 views

Having two random generated natural numbers between 1 and 255, and generate out of it natural number between 1 and 256

Let's say you have two cube with 255 sides and you have to use them to simulate a single cube with 256 sides, how can I do it? $f(n)$ and $g(n)$ returns random number between 1 and 255. I thought ...
2
votes
0answers
34 views
+50

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
0
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1answer
25 views

Pearson coefficient and rate of change [on hold]

Given that two series $(x_1,....x_n)$ and $(y_1,....y_n)$ are linearly correlated How can I measure the change in the increment ($i\leq j$ ) $y_j−y_i$ as a function of $x_j−x_i$ , Pearson's r and ...
0
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0answers
32 views

How can two random variables are continuous infers that their jointly random variable is continuous?

We assume that $\forall a,b$ such that $a2+b2>0$, $aX+bY $ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Under which ...
3
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0answers
34 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
1
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1answer
11 views

Question regarding the density function of first n prediction

This is an example from Bertsekas' Introduction to Probability 2nd edition example 8.2 Consider now a variation involving the first $n$ dates. Assume that Juliet is late by random amounts $$X_1, ...
1
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0answers
26 views

A 'mix' of simple and lazy simple random walk

Consider a $\mathbb{Z}$ valued markov chain $X_n$ which evolves as follows. $$P(X_{n+1}=y | X_n) =\begin{cases} \frac{1}{2}, y=X_n+1, X_n-1, |X_n|>K \\ \frac{1}{4}, y = X_n-1 , y= X_n+1, ...
1
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2answers
36 views

Probability Runs

Paper is produced in a continuous process. Suppose that a brightness measurement is made on the paper once every half hour. We have fifteen measurements. The data was entered into R and the following ...
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1answer
23 views

counterexample to conditional expectation

Let F,G be some $\sigma-algebra$ is it true that in general $E\left(E\left(X\mid G\right)\mid F\right)\neq E\left(X\mid F\cap G\right)$? I think it's not, however I can't provide a counter example
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1answer
51 views

Counting Probability [closed]

Consider the following equation $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 20$$ a) Count the number of integer solutions of the equation under the condition that $x_i \ge 0$ for $i=1, 2,\ldots 6$. ...
2
votes
1answer
33 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)log(n+1)}$$ Prove that $X_n$ ...
1
vote
1answer
91 views

Stationary Markov process properties

Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$, defined on $(\Omega, \mathcal{F}_t,P)$. Suppose that $X$ has stationary, independent increments. I now want to show the ...
2
votes
0answers
21 views

Proving that a local martingale given by a stochastic integral is not a martingale

Let $X_t=\int_0^t e^{W_s^2}dW_s$ for $0\leq t\leq 1$ and show that is not a martingale. I guess the reason is that the expectation is not finite, but I'm not sure how to show it precisely. In fact ...
1
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1answer
23 views

Are Markov chains necessarily time-homogeneous?

I've seen a definition of Markov chains as a stochastic process $(X_t)_{t\in I}$ fulfilling the weak Markov property and having index set $I = \mathbb{N}_0$. But the weak Markov property ...
0
votes
0answers
7 views

Logarithm of Brownian motion which is a local martingale but not a martingale

Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions starting at positive points (not necessarily at the same point). Let $X_t=\log(W_1^2+W_2^2)$ and show that it is a local martingale but not a ...
10
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3answers
816 views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
1
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2answers
24 views

Derivation of the Hypergeometric Distribution

The derivation of the hypergeometric refers to the following example: An urn contains N white balls, M black balls and we draw $n\le N+M$ balls without replacement. Let X be the number of white balls, ...
0
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0answers
19 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
1
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1answer
27 views

Measure Theoretic Definition of a Random Variable

I am struggling a little with the definition of a RV: Let $(\Omega,F),(\Omega',F')$ be two event spaces. Then every mapping: $X:\Omega \to\Omega'$ is a RV provided $X^{-1}A' \in F,~~~ \forall A' \in ...
0
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0answers
19 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 1)

I am quite new in the world of statistics, hence I am quite unsure when working with probabilities. I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a ...
0
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0answers
9 views

Chernoff type bounds for negative binomial distribution

If I recall correctly I remember reading that we cannot get Chernoff type results for the negative binomial distribution because of something regarding lebesque measure. I don't quite know all the ...
1
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0answers
8 views

Uniform Probability and Riemann Sum

Quoted from the Wikipedia page about Natural Density: We see that this notion can be understood as a kind of probability of choosing a number, which obviously is the reason why Natural Densities are ...
1
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0answers
10 views

The name of invariance principle of Donsker

I have seen the invariance principle of Donsker for the Wiener measure in Karatzas' Brownian Motion and Stochastic Calculus. I am wondering why this theorem have this name, e.g. where does the ...
2
votes
1answer
38 views

How many tosses are necessary that $n$ players produce pairwise different numbers of tosses with the result “heads”?

$n$ players toss a fair coin. The number of tosses with the result "heads" is recognized for each player. The game stops if the numbers are pairwise different. Let $X$ be the number of tosses for ...
1
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1answer
25 views

Expectancy of a joint density

A machine consists of two components, whose life times have the joint density function $ f(x,y)= \begin{cases} 1/50, & \text{for }x>0,y>0,x+y<10 \\ 0, & \text{otherwise} \end{cases} ...
4
votes
2answers
45 views

Rolling two dice… [closed]

Let $A_n$ be the number of fives, $B_n$ the number of sixes and $C_n$ the number of eights in $n$ rolls of two dices. For which n do we have: $E(A_n) < E(min(B_n,C_n))$ ?
1
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4answers
55 views

A Stochastic Limiting Inequality Proof

Let $(X_p)_{p\ge 0}$ be a sequence of non-negative random variables with finite mean for each $p\ge 0$. Then $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
1
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0answers
17 views

Determine if these are Characteristic functions

I'm having trouble determining if: $$\frac{sin(t^2)}{t^2}$$ and: $$\frac{1}{2} + \frac{1}{2}cos(t)$$ are Characteristic Functions. The first one I believe is very close to the sum of 2 uniformly ...
0
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1answer
22 views

How do you count probability from expected value and spread of a discrete variable? [closed]

One garden produces 500kg of fruit on avarage. Another produces 300kg on avarage. (Both yearly.) Their spread is 100kg and 80kg, and their correlation is 0,7. The expected value of both is 800kg. What ...
0
votes
1answer
19 views

if the probability a shooter hits the target is equal to .8 then …?

if the probability a shooter hits the target is equal to .8 then the probability that the shooter will correctly hit the target after 10 failed attempt is equal ......? probability of hitting the ...
-1
votes
0answers
18 views

Show that if $P(Y_1 \neq 0) > 0$, then with probability one, $\limsup\limits_{n} X_n = +\infty$, $\hspace{10mm}\liminf\limits_{n} X_n = - \infty$ [closed]

I'm having quite a bit of difficulty with the following problem. Any answer or detailed explanation would be greatly appreciated. Let $Y_1, Y_2, ...$ be bounded iid random variables such that $EY_1 ...
2
votes
2answers
41 views

Why is an entropy of $\text{log}(n)$ only compatible with the uniform distribution

I have a random variable $X$ and want to show that having an entropy $$ H(X) = - \sum_{i=1}^n p_i \text{log}(p_i) = \text{log}(n)$$ is equivalent to the distribution of $X$ being uniform. Starting ...
0
votes
1answer
24 views

Differentiating social surplus function

Can someone possibly explain how to >>make sense<< of the following identity: $\int \frac{\partial \ max_d \{ u(x,d) + \epsilon(d) \} }{\partial u(x,d)} q(d\epsilon \lvert x) = \int I\{d = ...
0
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1answer
25 views

Cauchy iid random variables and Strong Law of Large Numbers (helping understand)

Question: Let $(X_n)$ be a sequence of i.i.d. Cauchy random variables with density $\frac{1}{ π(1+x^2)}$. Use the characteristic function $φ(t) = e^{−|t|}$ of the Cauchy distribution to find the ...
3
votes
2answers
30 views

Notation $E[t^X]$ where $X$ is a random variable

I have a quick question which occured in the context of probability-generating functions but maybe the issue is more basic. For a random variable $X$, the probability-generating function is given as ...
2
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0answers
21 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
0
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1answer
13 views

Inf is not a stopping time in general

If ${\tau_n}$ , $n=1,2,3...$ are stopping times to a given filtration $F_t$, why in general it's not true to claim that $\inf_n {\tau_n}$ is a stopping time also? Thanks
0
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0answers
21 views

Risk-neutral (i.e. martingale) measure if density is given for a single random variable (i.e. asset)

Let $(\Omega,\mathcal F, P)$ be a probability space. And let $S : \Omega \to \mathbb R$ be a random variable, called an asset, also we are given $\pi > 0$ called a price and some $r \ge 0$ called ...