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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3
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1answer
451 views

Expected value of a log Poisson distribution

Suppose there is a sequence $(X_n)_n$ of independent random variables, $X_i \sim Poisson(\lambda)$. I order to almost surely compute $\lim\limits_{n\to \infty} \sqrt[n]{X_1X_2\dots X_n}$, I thought ...
1
vote
1answer
19k views

How to find the the probability using mean and standard deviation

The IQs of $9$ randomly selected people are recorded. Let $\overline{Y}$ denote their average. Assuming the distribution from which the $Y_i$'s were drawn is normal with a mean of $100$ and a standard ...
6
votes
0answers
198 views

Question on Conditional expectation

Let $X_1$ and $X_2$ be two random variables on $(\Omega,\mathcal{B},P)$. Suppose there is a function $g:\mathcal{B}\times\mathbb{R}\rightarrow[0,1]$ such that for any $x$, $g(\cdot,x)$ is a ...
3
votes
1answer
56 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
2
votes
1answer
141 views

Mean of iid random variables, problem understanding a passage in a paper

I am presently reading and studying a paper (page 11 of this document) regarding epidemiology and disease spread over a network of contacts. I am having problems understanding a passage. Some ...
7
votes
4answers
328 views

How variance is defined?

The variance of a random variable $X$ is defined as $E[(x-\mu )^2]$. Why can't it be defined as $E[|x-\mu |]$. i.e., What is the basic idea behind this definition. Thank you.
2
votes
1answer
289 views

Convergence in distribution and convergence in the vague topology

From Terrence Tao's blog Exercise 23 (Implications and equivalences) Let ${X_n, X}$ be random variables taking values in a ${\sigma}$-compact metric space ${R}$. (ii) Show that if ${X_n}$ ...
2
votes
1answer
320 views

Counter-example for Fatou's lemma using a probability measure

Fatous's lemma states that: Let $f_1,f_2, \ldots, f $ be Borel Measurable and $f_n \leq f$ for all n, where $\int_\Omega f \;d\mu < \infty$. Then $$ \limsup_{n\rightarrow\infty} \int_\Omega f_n ...
1
vote
1answer
84 views

How is $P(|A) $ defined from $P(|\mathcal{B})$?

Let $ (\Omega, \mathcal A, \operatorname{P})$ be a probability space, with a real random variable $X$ and a sub-σ-algebra $ \mathcal B \subseteq \mathcal A$ In probability theory with measure ...
6
votes
1answer
921 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
4
votes
2answers
385 views

Using Recursion to Solve Coupon Collector

I read a brilliant answer by Mike Spivey on one of the questions and I was wondering how I could use it to solve a coupon collectors problem. The problem is : There are coupons labelled 1,2,3...,10 ...
12
votes
1answer
326 views

pointwise limit of finite measures

If there is a sequence of measures $\mu_n$ such that $\mu_n(A) \overset{n}{\rightarrow} \mu(A)$ for all $A$ in the $\sigma$-field and if $\mu_n(\Omega)\leq c$ $(c<\infty)$ for all $n$, then $\mu$ ...
3
votes
1answer
511 views

Bayesian Inference in Measure Theory

What's the deal. How does this work, or can you point me to some references? I tried $\mu(A|B) = \mu(A \cap B) / \mu(B)$ and got stuck on $\mu(B) = 0$. Edit: Sorry for being lazy. My background is ...
3
votes
3answers
558 views

Prove the Probability of Two Events

I'm taking my first course in Probability and one of my homework problems is to prove that for any two sets: $$P(A \cup B) = P(A) + P(B) - P(A \cap B) $$ Note that the function $P$ is the ...
1
vote
1answer
38 views

What is the conditional $f_{X|Y}$ and expectation of $X|Y$ over a irregular area $f_{X,Y}(x,y)=.01$

I am confused as to how to integrate, would one split this up into sections ?
19
votes
2answers
763 views

Cover time chess board (king)

Consider a random walk of a king on a standard chess board, which at each step moves to a uniformly random permitted square. What's the exact mean time to visit all squares (cover time), starting ...
2
votes
2answers
251 views

How is independence between two family of random variables defined?

Given a family of random variables are jointly independent, how is independence between two family of random variables defined? An example is from Terry Tao's blog. Thanks! Exercise 18 (Creation ...
0
votes
1answer
48 views

interpreting an expression involving two random variables

Consider a function $$g=E[\max(a+X,d+Y)]$$ where $a,d\in R$ and $X$ and $Y$ are independent and identically distributed standardized random variables with mean $\mu$, variance $\sigma^2$, continuous ...
0
votes
2answers
179 views

Probability that total weight of coffee in three 10-ounce jars is greater than the weight in one 30-ounce jar.

Suppose that instant coffee comes in two sizes, 10-ounce jars and 30-ounce jars. Let $X$ be the actual weight of coffee in a 10-ounce jar and assume that $X$ has a normal distribution with a mean ...
4
votes
1answer
445 views

Radon-Nikodym - Martingale

I have a question concerning point 3 of the following problem: Let $(\Omega,\mathfrak{F},\mathbb{P},\mathfrak{(F_n})_{n\in\mathbb{N}})$ be a filtered probability space and $\mu$ a finite measure on ...
2
votes
0answers
61 views

Pointwise limit of measures [duplicate]

Possible Duplicate: Limit of measures is again a measure If $$ P(A) = \lim_n P_n(A)$$ for every $A \in$ $\sigma-$field $\mathcal{F}$. Can we prove that if $P_n$ are probability measures ...
3
votes
2answers
115 views

How is this done by dominated convergence theorem?

Terry Tao wrote in his blog Fix ${k \geq 1}$. If ${X}$ has finite ${k^{th}}$ moment, say ${{\bf E}|X|^k \leq C}$, then from Markov’s inequality (14) one has $$ \displaystyle {\bf P}(|X| \geq ...
0
votes
3answers
107 views

Equality of two probabilities

I would like to know what should I verifiy in order to show that two probabilities are equal. Here is the exercice : Let $F_0$ be an algebra of sets over $\Omega$ and $P$, $P'$ two probabilities ...
1
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0answers
78 views

Inequality with Expectation and vectors

Let $x=(x_1,\ldots, x_n)\in R^n$. Let $r_i, i=1, \ldots, n$ be Rademacher i.i.d. random variables (i.e. $P(r_i=1)=P(r_i=-1)=1/2$). It is a well-known inequality that: $$ E\left(\left|\sum_{i=1}^n ...
11
votes
1answer
326 views

What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia) For some sets, the ...
0
votes
2answers
108 views

Calculate probability of many events

I was wondering if there is a simpler or more straight forward way to go about solving probablity questions that envolve many events, without having to make gigantic tree diagrams or tables. For ...
1
vote
0answers
89 views

Principle of single big jump

I'm new here, i know that from the principle of single big jumps it follows, $$\Pr[X_1+ \cdots +X_n>x] \sim \Pr[\max(X_1, \ldots,X_n)>x] \quad \text{as } x \to \infty. $$ So let $F$ be a ...
2
votes
1answer
2k views

Division of two random variables of uniform distributions

Having X ~ Uniform(0,1), Y ~ Uniform(1,3) independent what's the pdf of Z = X/Y. This means I can write the PDFs as follows $$f_X(x) = 1$$ for $ x \in \left(0,1\right)$ and 0 otherwise $$f_Y(y) = ...
0
votes
1answer
186 views

Random walk with zero drift

Let $X_{k+1} = X_k+\xi_k$ be a random walk on $\Bbb R$ starting from $0$ and such that $\mathsf E\xi_0 = 0$, $\mathsf{Var}[\xi_0]>0$. Is that true that $$ -\infty = ...
1
vote
1answer
101 views

One variation of Monty Hall problem

Here once Monty has a choice to open any of two doors each containing a goat (i.e. when our initial choice is a car) then he chooses the rightmost one. We need to prove that here also the "overall" ...
10
votes
1answer
444 views

Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
4
votes
1answer
206 views

CLT for random variables with heavy tails

The simplest form of classic CLT can be put in the following form: If $X_1,X_2,\dots$ are i.i.d. with mean $0$ and variance $1$, then the distribution of the normalized sum ...
2
votes
2answers
796 views

Why $\sigma$-algebras represent information, and what information does $\sigma(X)$ represent?

I am confused about the notion of $\sigma$-algebras representing information and what information is contained in $\sigma(X)$ for a random variable $X$. Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is ...
1
vote
1answer
206 views

Question regarding distributions of min/max functions

I am having trouble with the following problem and was wondering if someone can point me in the right direction. Let $X_1, X_2, \ldots$ be an infinite sequence of independent, identically distributed ...
0
votes
1answer
133 views

Can I use $P(x_1+x_2+…+x_n<k)$ to calculate $P(y_1+x_2+…+x_n<k)$?

Suppose that we have $n$ independent Bernoulli random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value 1 with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ...
6
votes
2answers
109 views

Closeness of probability measures

Consider a set of probability measures $\{P_n\}$. Suppose $P_n$ converges to $P^*$ weakly and $$ \int \xi^2 P_n(d\xi)< \infty. $$ Can we claim $$ \int \xi^2 P^*(d\xi)<\infty $$ and $$ \lim_{n\to ...
2
votes
1answer
138 views

Mathematical expectation is inside convex hull of support

Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$? Let $s(x)$ be a support function of set ...
1
vote
1answer
38 views

Distribution of $\int X_t \,dt$ if we know distribution of $X_t$

If $B(t)$ is the standard Wiener process on [0,1], how should one go about finding the distribution of some random variable $Z$ defined by, say $$ Z(\omega) = \int_0^1B_t(\omega)\,dt\quad ?$$ ...
3
votes
0answers
125 views

Relation between maximizer's derivative and maximizing function

Let $u(x)$ be a continous bounded function such that $u'(x) >0$, $u''(x) < 0$. Define $A(x) = -\frac{u''(x)}{u'(x)}>0$. Let $Y$ be a random variable with $\mathbb{E}|Y|<\infty$. I solve a ...
0
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0answers
139 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 ...
0
votes
1answer
43 views

independence of uniform random variables1

let $X_j \sim U(0,1)$ if $$Y_j=\frac{X_j}{X_1+X_2+\cdots+X_n}$$ I want to show that: $Y_j $are independent $\operatorname{Var}(Y_1)=\dfrac{c}{n^2} +o\left(\dfrac{1}{n^2}\right)$ then calculate ...
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0answers
32 views

$dt=dX_t=d\bar X_t$ on the event $\{X_t = \bar X_t, \bar X_t > w \}$, $\bar X$ running supremum of $X$

Let $X_t$ be a Levy process, with its associated running supremum $$ \bar X_t = \sup_{s \le t} X_s. $$ With $w >0$, define the process $$ W_t = \left( w \lor \bar X_t \right) - X_t. $$ Show ...
4
votes
0answers
83 views

The limit in law of a sequence of normal distributions is normal [duplicate]

Let $ \{ \xi_n \}_{n=1}^{\infty}$ be a sequence of normal random variables, where $ \xi_n\sim\mathcal{N}(\alpha_n, \sigma_n^2)$ and $\xi_n \overset{d}{\rightarrow} \xi$. I need to prove, that $\xi$ is ...
3
votes
2answers
1k views

What is a sampling density? Why is the sampling density proportional to $N^{\frac{1}{p}}$?

I'm reading a book named The Elements of Statistical Learning by Hastie, in section 2.5, Local Methods in High Dimensions, it says that the sampling density is proportional to $N^{\frac{1}{p}}$, where ...
1
vote
1answer
240 views

Is this Markov?

Consider a process $\{X_n, n\geq 0\}$ with state space $S=\{0,1,2\}$ s.t. $$ P(X_{n+1}=j | X_n=i, X_{n-1}=i_{n-1}, \dots, X_0=i_0)=\begin{cases} P_{ij}^I \ \ \ n \ \mbox{ even},\\ P_{ij}^{II} \ \ \ ...
1
vote
1answer
33 views

Show that functions are affinely dependent

Let $u(x),v(x)$ be continuous bounded functions on $\mathbb{R}$ such that for any Borel probability measures $\mathbb{P}_{1},\mathbb{P}_2$ on $\mathbb{R}$ $$ \int u(x) \, \mathbb{P}_1(dx) \leqslant ...
3
votes
2answers
257 views

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
1
vote
1answer
120 views

Lévy-Khintchine formula for Cauchy distribution

The Lévy-Khintchine formula for the log of the characteristic function of an infinitely divisible random variable is $$ \Psi(s)=ias + \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} + ...
3
votes
1answer
242 views

Fixed-Time Brownian Motion Exit Probabilities

A standard computation using martingale techniques allows us to compute probability that a Brownian motion started at zero exits the interval $[-a,b]$ ($a, b > 0$) at $-a$ or $b$. It appears to me ...
1
vote
3answers
422 views

Is this function - characteristic function of a random variable?

$\phi(t)= \begin{cases} 1,&\text{if $|x|< 1$;}\\ e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.} \end{cases} $ Can anyone help?