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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149 views

question about conditional probability and $\sigma$-algebra

I have a question: Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and two random variables $X$ and $Y$. For a Borel-measurable set $\Gamma$, if there exists a measurable function $g$ ...
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54 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
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2answers
256 views

Conditional expectation (mixed with an iterated expectation) $E[E(X\mid Y)\mid Y]=E(X\mid Y)$

Conditional expectation: I want to prove $E[E(X\mid Y)|Y]=E(X\mid Y)$ I attempted the following. Is it correct? $$\begin{align*}E[E(X\mid Y)|Y=y]&=\int_{-\infty}^\infty E(X\mid Y=y)f_{X\mid ...
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1answer
383 views

Eigenvector of transition matrix for Markov chain

Why is the only eigenvector of the transition matrix for an irreducible Markov chain with eigenvalue $= 1$ the eigenvector with all ones?
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2answers
90 views

Need help with derivation of conditional expectation

The following is taken from the book "Mathematical Statistics for Economics and Business": \begin{align*} E\left.\left( \left[ Y-h(x) \right]^2\ \right\vert\ x\right) =& E\left.\left( ...
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0answers
94 views

On continuity of measure

Let $m$ be a probability measure on $\mathbb{R}^n$. Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$. Say under what conditions the following inequality holds. $$ m\left(\left\{ x \in ...
3
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1answer
135 views

Question about a separation theorem

This theorem is called the Kreps-Yan theorem. I have just a small question about the proof. We have a probability space given. The statement is: Let $p\in [1,\infty]$, $q$ the conjugate. Suppose ...
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1answer
30 views

Is it possible to reverse probabilistic automaton?

Is it possible to reverse probabilistic automaton (PA), i.e. calculate the probability of previous state given current state? Will reversed automaton be a PA (Markov?), i.e. will next probability ...
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2answers
398 views

Snell envelope and optimal stopping time

Suppose $(G_n)_{0\leq n\leq N}$ is a process adapted to a filteing $(\mathcal{F}_n)_{0\leq n\leq N}$. The Snell envelope of $(G_n)$ is the smallest supermatingale dominates $(G_n)$. It's defined as ...
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2answers
683 views

Linearity of conditional expectation (proof for n joint random variables)

Linearity of conditional expectation: I want to prove $$E\left(\sum_{i=1}^n a_i X_i|Y=y\right)=\sum_{i=1}^n a_i~ E(X_i|Y=y)$$ where $X_i, Y$ are random variables and $a_i \in \mathbb{R}$. I tried ...
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1answer
49 views

Probability solution verification

On a statistics trial exam I encountered the following tricky exercise: Assume that there are two types of car drivers in a country. Safe drivers constitute $70$% of the population and they have a ...
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2answers
79 views

What can be said about $E[1_A\mid\mathcal F]$?

It is known that $E[X\mid 1_A]$ is of particularly nice form. What can be said about the form of $E[1_A\mid\mathcal{F}]$ for "general" $\mathcal{F}$? Is it true that $E[1_A\mid\mathcal{F}]=1_B$ for ...
1
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1answer
127 views

Uniform convergence in probability

Let $X_i$ and $Y_i$ $(i\in\mathbb{N})$ be two strictly stationary sequences of real valued random variables with finite variance. We have two empirical distribution functions $F_{1,n}:=\frac{1}{n} ...
1
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1answer
94 views

Gebelein's Inequality and convergence of distribution

We know that for a bivariate standard normal vector $Z=(Z_1,Z_2)$ it holds that \begin{align*} \operatorname{Cov}(1\{Z_1\leq u),1\{Z_2\leq u))\leq \operatorname{Cov}(Z_1,Z_2). \end{align*} This ...
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0answers
82 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
6
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1answer
320 views

Equality of a supremum of two Expectation

I got stuck to prove an equality in a detailed way and I hope, someone could tell to me, how to fix it. Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $\mathcal{K}$ be the set of all ...
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1answer
60 views

Writing the definition of the expectation $E(X+Y \mid Z=z)$

If I want to write out the "definition" of the conditional expectation $E[X+Y \mid Z=z]$, would it be (for the continuous case): $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x+y)\,f_{X,Y\mid ...
3
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1answer
362 views

Geometric distribution with unequal probabilities for trials

I am researching an engineering problem in which I want to model the probability distribution of the number X of independent trials needed to get one success. If the probability of success at each ...
5
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2answers
261 views

Proof of the infinitude of primes by probabilistic methods.

I'm looking if there is proof of the infinitude of prime numbers using probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between ...
1
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1answer
79 views

Supremum over dense subset

I'm interested about the following supremum: $$\sup_{g\in A}E[-gf]$$ where $A\subset\{g\in L^1:g\ge 0,E[g]=1\}$ and $f\in L^\infty$ is fixed. $E$ denotes the expectation, i.e. it is the integral ...
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0answers
32 views

Probability of a sum being above a threshold in a n dimensional vector

I am having a problem measuring a certain probability. Suppose a n-dimensional vector a1,a2,...,an. What are P(a1+a2+...+an < 1) , P(a1+a2+...+an = 1) and P(a1+a2+...+an > 1) equal to? ais are all ...
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2answers
45 views

if $X\sim U(0,1)$ show that $(b-a)X+a \sim U(a,b)$

I'm using the MGF method, this is what I get: $$ \begin{align} Y&=(b-a)X+a\\ M_Y(t)&=E[e^{(b-a)X}e^a] \\ &=E[e^{(b-a)X]}e^a &\text{I think this is my error} \\ &= M_x((b-a)t)e^a\\ ...
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0answers
70 views

A generalization for martingales?

I am a computer scientist interested in analyzing stochastic processes specified as probabilistic programs. In my research, I recently encountered an idea that looks just like a martingale, but is ...
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2answers
30 views

Expectation with a “regular” function

I hope this is not a silly question. I know that the expectation of a constant is just a constant (i.e. $E[c]=c$ for $c\in \mathbb{R}$), and that for a function $g$ of a random variable X, ...
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2answers
211 views

How many ways to draw consecutive fibonacci numbers from deck of cards

In a deck of cards there are 4 suits of 13 cards each. If the face value of the aces is defined as 1 and the jack, queen, and king are 11, 12, and 13 respectively, then: 1) What is the probability ...
3
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1answer
440 views

Physical meaning of “probability density”

Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. ...
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2answers
156 views

That Brownian Motion's increments are gaussian is “not surprising”?

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ...
1
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0answers
137 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
3
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1answer
156 views

Can I identify the distribution of a random variable given a related distribution function?

Let $X_1$, $X_2$ be i.i.d random variables. Suppose we know the distribution function of $X:=|X_1-X_2| =\max\{X_1,X_2\} - \min\{X_1,X_2\}$. Can we find the distribution of $X_1$? I realize that ...
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1answer
250 views

Jensen inequality

Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random ...
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2answers
61 views

Why do we care about specifying events in a probability space?

Why aren't probability spaces just defined as $(\Omega, p)$ pairs with $\Omega$ as the sample space, $\sum_{\omega \in \Omega}p(\omega) = 1$, and for a subset $A \subseteq \Omega$, $\Pr(A) := ...
3
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3answers
228 views

Derivative of the maximum of two random variables

For any two real numbers $a$ and $b$ and any two random variables (with no mass points in their distributions) $x$ and $y$, why is it that the derivative of $E[\max\{a+x,b+y\}]$ with respect to $a$ is ...
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3answers
78 views

I do not understand this integral,please help…

$$\int_0^{\infty} P(y > z) \, dz = \int_0^{\infty} \int_z^{\infty} h(y) \, dy \, dz = \int_0^{\infty} \int_0^y \, dz \, h(y) \, dy$$ Why do we have the last equality? I used Fubini and derived the ...
4
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1answer
205 views

Understanding the definition of a random variable

I'm working through a math stats book on my own (I've always wanted to learn it), but I'm getting confused about the definition of a random variable. The book says that a random variable is a ...
2
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1answer
65 views

Classical probability, need my work checked.

A kinder egg may contain a prize of type $A$, of type $B$, or be empty. The probability of a prize $A$ is $5$ times larger than of prize $B$. A box contains 12 kinder eggs, from which 7 are known to ...
4
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1answer
479 views

Convergence in distribution of Gaussian processes

Assume given a sequence $(W_n)$ of Gaussian processes indexed by, say, $\mathbb{R}^p$, with mean zero and covariance function $R_n$. This means that for each $n$, the finite-dimensional distributions ...
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2answers
185 views

Lower bound on the probability that the maximum of a sequence of $n$ i.i.d. standard normal r.v.'s exceeds $x$

Let $X_{\max}=\max(X_1,X_2,\ldots,X_n)$ where $n$ is large and each $X_i$ is i.i.d. standard normal random variable, i.e. $X_i\sim\mathcal{N}(0,1)$. Is there a lower bound on the probability ...
2
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2answers
477 views

An Orlicz norm is a norm

I had asked a question pertaining to Orlicz norms here. However, in the book I was reading, it said (and I paraphrase) "It's not difficult to show it is a norm on the space of integrable random ...
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1answer
32 views

Question regarding GWP (Kesten-Stigum setup)

Let $(Z_n)_{n\in \mathbb{N}}$ be a GWP with $Z_0=1$ and mean of offspring distribution $m\in (1,\infty)$. Define $W_n=Z_n/m^n$ and denote its limit by $W$ (i.e. setup as in the Kesten-Stigum theorem). ...
1
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1answer
83 views

How to prove this simple inequality?

Please help me to prove this inequality. Suppose $X$ and $Y$ are independent and $EX=EY=0$, then we must have $E(|X|) \leq E(|X+Y|)$. Thanks.
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1answer
50 views

Convergence of expression involving normal c.d.f.

I have derived the following expression for the error of some approximation: for any $\epsilon > 0$ $|u_{precise} - u_{approx}| \leq C_1 \epsilon + C_2 \cdot \Phi \left ( -\sqrt{-2 \ln ...
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1answer
172 views

Convergence w.p. 1 vs convergence in probability: a “physical” example

I understand (proved) that convergence with probability one implies convergence in probability, and that the latter notion is indeed weaker; I've completed an exercise showing that a sequence of ...
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0answers
219 views

Hitting probabilities for Brownian motion

Let $\mathbb D$ be the complex unit disk. Let $B$ be a standard complex Brownian motion started at $0\in \mathbb D$. Let $\tau = \inf\{ t : B_t \in \partial\mathbb D\}$. I am trying to show that if ...
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1answer
72 views

What is the meaning of this exercise?

The daily quantity demanded of unleaded gasoline in a regional market can be represented as Q=100-10p+E, where p belongs to [0,8], and E is a random variable having a probability density given by ...
3
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2answers
653 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: ...
1
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1answer
91 views

Characteristic function of series

It is well known that if $X,Y$ are independent random variables on $(\Omega,\mathscr{F},P)$ with respective characteristic functions $\varphi_X,\varphi_Y$, then $\varphi_{X+Y}=\varphi_X\varphi_Y$. If ...
2
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1answer
123 views

Series of continuous random variables is continuous

We work on the usual $(\Omega,\mathscr{F},P)$. Suppose $X_i$ are independent random variables. Say the distribution of $X_i$ is $F_i$. Under what circumstances can I guarantee that $\sum_{i=1}^\infty ...
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0answers
25 views

ML Estimation for number of animals in a park. Hypothesis Testing.

A park of area $S=10 000 km^2$ was surveyed for bears, and out of $n$ disjoint regions of equal area $s=1km^2$, there were $n_k$ regions with $k=0,1,....,N$ bears. On each of these regions, the amount ...
2
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1answer
211 views

Unexpected Probability Theory Uses

I am a french student in mathematical engineering. I had to go trough an intensive 3 year "preparation" to pass a "concours" to go to High School. In mathematics, I have been taught a lot of algebra, ...
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0answers
31 views

Distributional convergence question for Feller processes

To briefly go over the setup, $S$ the state space is a separable locally compact metric space, and $C_0$ is the space of continuous functions on $S$ that vanish at infinity. $D$ is the space of ...