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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Why does a time-homogeneous Markov process possess the Markov property?

Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
3
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1answer
111 views

Product Measures

Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = ...
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4answers
161 views

Conjunction fallacy

I was reading this article which has the following question, Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with ...
3
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1answer
578 views

Relations between Order Statistics of Uniform RVs and Exponential RVs

Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
3
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2answers
360 views

Expectation of a stopping time uniquely determined by a function

Let $(X_t)_{t\ge0}$ be a Markov chain on a finite state space $\Omega$, with transition probability $P$. Let $T$ be a stopping time such that $T=\min \{t\ge 0;X_t \in A \subset \Omega \}$.  If ...
3
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1answer
159 views

Relation: pairwise and mutually

Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions ...
3
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2answers
150 views

probability -Diverging expectation

As I keep reading probability books, there are always some issues that no one considers. For example, for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega ...
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2answers
106 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
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2answers
154 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
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2answers
39 views

$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y$

$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y.$ Solution. $$f_{Y|X}(y|x) = \frac{1}{x^2}, \text{ $x \in (0,1]$, $y \in \mathbb{R}$.}$$ Thus $$f_{X,Y}(x,y) = ...
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2answers
206 views

Distribution of sums

I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that ...
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5answers
1k views

Some case when the central limit theorem fails

If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite ...
2
votes
2answers
122 views

sub martingales and more

This is a problem on sub-martingales. Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and where $S_n$ is a symmetric random walk and $\mu$ is greater than zero. We ...
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3answers
380 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
2
votes
2answers
247 views

Detail in Conditional expectation on more than one random variable

I have $E(X|Y,Z)=0$, $X$ independent of $Y$ and of $Z$ and I want to conclude that $E(X)=0$ ($X,Y,Z$ are real-valued random variables). Okay it seems quite obvious, but if I try to make a strict ...
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4answers
1k views

Why isn't there a uniform probability distribution over the positive real numbers?

Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what ...
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0answers
92 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
32 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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1answer
31 views

Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$

We need to show that, given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, that $P(C\mid A)≥1-\varepsilon$. Since we know that $P(C\mid B)=1$, it follows that $P(B ...
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1answer
63 views

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds:

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds: $P(Z\leq -u \space \text{or} \space Z\geq v) \leq \frac{4\sigma^{2} + ...
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vote
1answer
232 views

How can I show that the conditional expectation E(X|X)=X?

I tried to show that $E(X|X=x)=x$, which would lead me to get $E(X|X)=X$ but I am having trouble doing so. I know that the definition of conditional expectation (continuous case) is: ...
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0answers
52 views

What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.

In my text, there is a passage that says: "Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is: $$ \begin{align*} f_{s\mid ...
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1answer
78 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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0answers
275 views

complex integration over the whole plane

I am trying to solve this integral: $H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$, where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
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1answer
318 views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
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1answer
120 views

Continuity of Expected Value

Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$. Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, ...
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1answer
437 views

Prove that vector has normal distribution

You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$. Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$. You build following random variables, based ...
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1answer
426 views

recursive equation for number of white balls

Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls ...
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55 views

Probability Theory - Fair dice

Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. Im unsure ofwhat type of question this is, I have tried ...
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0answers
43 views

chose uniformly at random from the n different brands, independently of previous orders [duplicate]

Michiel's Craft Beer Company (MCBC) sells $n$ different brands of India Pale Ale (IPA). When you place an order, MCBC sends you one bottle of IPA, chosen uniformly at random from the $n$ different ...
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1answer
32 views

A basic question on expectation of distribution composed random variables

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$. If $F$ and $G$ have no common jumps then I need to show that $E[F(Y)] + E[G(X)] = 1$. How to proceed here ? ...
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2answers
28 views

PDF of the addition of several outcomes from Poisson distribution

We draw $n$ values from a Poisson distribution and add them. - What is the expected of this addition - What is the PDF of this addition It seems quite intuitive to me that if we add $n$ Poisson ...
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1answer
55 views

Expected Value and Variance of Two Random Variable

Let $X_1, X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ be independent exponential distributed random samples with mean $\theta$. Let $T\alpha = \alpha\bar{x} + (1-\alpha)\bar{y}$, where $0 < \alpha < 1$. ...
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0answers
50 views

Transformation of Multiple Variable

I'm having trouble with the following exercise: Let $Y_1, Y_2, \dots, Y_n \overset{\rm i.i.d.}\sim \exp(\theta)$ are random samples. If $Y_i$'s are sorted in ascending order, the ordered random ...
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1answer
59 views

Approximate Expectation of $x^2$

I am estimating the $E[x^2]$. The function I am looking at gives you a constant value of A from 0 to 4, it is 0 from 4 to 6, and A from 6 to 10. I got that $E[x]$ to be 5. I calculated the value of ...
0
votes
1answer
52 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
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0answers
55 views

Probability of convergence of a monotone sequence

Let $\left(\Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, and let $X: \Omega \rightarrow \mathbb{R}^n$ be a random variable. Let $\mathbb{P}^N$ denote the product measure $\mathbb{P} ...
0
votes
1answer
204 views

Expected value of c.d.f when normal distributed

I need help to calculate the expected value of an invertal of a c.d.f function which is normal distributed. I know that $E(X)=\int^\infty_0 (1-F(x))dx$ What i need is to calculate $E(w|w \geq ...
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votes
1answer
553 views

What is the PDF of a product of a continuous random variable and a discrete random variable?

Let $N$ be a discrete random variable which takes values in [0, ..., M], M > 0, with known PDF $P(N=n)$. Let also the continuous random variable $Z = \sum_{i=1}^{N}X_{i}$ as the sum of i.i.d. $X_{i}$ ...
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99 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
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1answer
197 views

Probability of two opposite events

Suppose there is string of eight bits, e.g.: 00100110 Bits are randomly chosen from the string. All choices are done equally likely. Probability of choosing $0$: $p_0 = \frac{5}{8} = 0.625$ ...
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2answers
489 views

Minkowski's Inequality For Infinity

I've tried figuring this out and searching the net on this for 5 hours, but I can't get it. Every source says it's trivial, but I must be missing something because I have pages of work that don't lead ...
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1answer
899 views

Finding an expression for the probability that one random variable is less than another, given a condition.

Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as: $$\mathbb{P}[X < Y] = ...
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1answer
736 views

pdf equation for tossing 2 coins given the probability of landing head for each coin in a single toss

Problem: Consider a simple coin-flipping experiment in which we are given a pair of coins A and B of unknown biases, $\theta_{A}$ and $\theta_{B}$ respectively (that is, on any given flip, coin A ...
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3answers
2k views

Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
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3answers
1k views

Random Variable Inequality

Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result: (A-S ...
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1answer
698 views

What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
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2answers
983 views

How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?

I'm learning Kolmogorov's zero-one law in probability theory: Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in ...
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1answer
270 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
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1answer
129 views

Is there one-tailed version of Vysochanskiï–Petunin inequality, like Chebyshev?

The Vysochanskiï–Petunin inequality gives a tighter bound than Chebyshev for unimodal distributions . I'm just wondering if there is a one tailed version of it, like that of Chebyshev inequality? ...