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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
24 views

If $P(X<Y)=P(X<g(Y))$ then what could be the form of $g$?

Let $X$ and $Y$ are two continuous random variable and $$P(X<Y)=P(X<g(Y)),$$ for some convex function $g$. Is it true that $g$ will always be a linear function?
1
vote
1answer
15 views

limit of average of independent, but not identically distributed r.v.

Let $\{X_i\}$ be a collection of independent r.v., but with distribution dependent on index $i$, such that $P(X_i=2^i)=2^{-i}$ and $P(X_i=0)=1-2^{-i}$ for $i \in \mathbb{N}$. What can I say about ...
0
votes
1answer
34 views

CLT version for $ER_n(p)$ graphs

We defined the Erdôs- Rényi graph as follows: $ER_n(P)$ is the random graph with vertex set $[n]$ where each pair $\{u,v\}$ of vertices is added to the edges set $E$ independently with probability ...
1
vote
0answers
20 views

How to prove that $cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0$ for $X_1, \ldots, X_n$ independent and $f,g$ increasing?

I read in a talk that a consequence of the FKG inequality is that: $$ cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0 $$ for $X_1, X_2, \ldots ,X_n$ independent and $f,g$ increasing ...
0
votes
1answer
20 views

Family of decompositions of a probability space and sigma algebra generated by a discrete random variable.

While reading the textbook "Martingale Methods in Financial Modelling" by Musiela and Rutkowski I am puzzled with a new definition (i.e. "the family of decompositions") that I never encountered and ...
0
votes
0answers
10 views

Problem with change of variables (with regard to kernel density estimation)

I'm trying to understand why \begin{equation*} \begin{split} E(R(\hat{f}'')) & = \frac{1}{nh^6} \int \int K'' \left( \frac{x-y}{h} \right)^2 f(y) dy dx \\ & \quad + \frac{n(n-1)}{n^2 h^6} ...
0
votes
0answers
7 views

Does this self-conjured RV converge almost surely?

I thought of this example in hopes of helping me understand almost sure convergence a little better. So, if you could add any additional (relevant) details in your response I would greatly appreciate ...
1
vote
0answers
32 views

Showing that $E[X|G_1,G_2]=E[X|G_1]$, where $\sigma(X,G_1)$ and $G_2$ are independent

Suppose that $X$ is an integrable r.v. on probability measure space $(\Omega,F,P)$. Show that $E[X|G_1,G_2]=E[X|G_1]$, where $G_1,G_2$ are sub $\sigma$-algebras and $\sigma(X,G_1)$ and $G_2$ are ...
11
votes
5answers
2k views

High school Math: confusion about the basic probability

I am confused about the following two scenarios: Out of a bag of 3 apples and 3 oranges, you pick 2 items. 1) What is the probability that you will have 2 apples? 2) What is the probability that ...
0
votes
0answers
34 views

Expectation and the Survival Function: Measure Theory

I have an example from class notes that I do not understand and would appreciate some clarification. Particularly, I haven't found a direct explanation online or in my text with regards to the limits ...
0
votes
1answer
12 views

Monotone convergence implies $\mathbb{E}\sum X_n = \sum \mathbb{E}X_n$?

My professor stated the following implication made from the Monotone Convergence Theorem: \begin{align}\mathbb{E}[\sum X_n] = \sum \mathbb{E}[X_n] \end{align} Up till now I have been assuming the ...
0
votes
3answers
48 views

If $P(X = x) = 0$ for a continuous RV $X$, then isn't that it is impossible to observe any data at all?

For example, if I let $X$ be the weight of a dog, then I weigh my dog and his weight is 10 lb. Theorem says the probability of observing this 10 lb is zero, i.e $P(X=10) = 0$. However, I DO observe ...
1
vote
1answer
33 views

When is an improper Riemann integral equal to Lebesgue integral

My original problem is given $X_i\sim^{iid}U[0,1]$, find $$\lim_{n \rightarrow \infty} (X_1X_2 \cdots X_n)^{1/n} = \lim_{n \rightarrow \infty} (\prod_{i=1}^{n} X_i)^{1/n}$$ Well, $$\lim_{n ...
1
vote
0answers
30 views

Weak convergence function

I am studying for an exam and this was one of the hard problems in my textbook. Let $f_n(x) = 1 −\cos(2\pi nx)$ for n $\in$ N and x $\in$ $[0,1]$. Verify that $f_n$ is the density of a probability ...
0
votes
1answer
47 views

Proving almost sure convergence

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ ...
1
vote
0answers
30 views

Convergence in distribution for continuous functions of random variables

If two sequences of random variables $\{X_n\},\{Y_n\}$ are such that $X_n \xrightarrow{d}X,\ Y_n \xrightarrow{d}Y $, i.e, they converge in distribution to $X$ and $Y$ respectively, where $X_n$ and ...
0
votes
0answers
30 views

First passage time of the Brownian motion

In an exercise (4.1 Krapinsky, "A kinetic view of statistical physics") I am asked to show that: The probability that a brownian motion on a 1D discrete lattice never reaches the site $n$ scales as ...
2
votes
2answers
48 views

Expectation Functional in Lebesgue and Riemann Terms – Looking for a clarification

Here there is a really central problem I am having self-studying probability theory, that concerns the relation between the definition of expectation in Lebesgue terms and in Riemann terms. I will ...
1
vote
3answers
45 views

Random variables and reference probability measure

I am self-studying probability theory, and I am having quite some problems with the very basic concepts of the theory that are seriously hampering any attempt to proceed further in my study. Here ...
0
votes
0answers
19 views

semimartingale and limit

Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g: \Bbb R \to [-1,1]$, of class $C^{\infty}$, with $g(x)= \left\{ \begin{matrix} -1, & x \le 0 \\ 1, & x \ge 1 \end{matrix} \right.$. ...
1
vote
1answer
70 views

Properties of a random walk [closed]

First of all, I know nothing about Markov chains, and I'd like to prove the following without using the theory around them. Let $(M_{n})_{n\geq 1}$ be a random walk over $\mathbb{Z}$, starting at ...
22
votes
2answers
702 views

Convergence of series $\sum\limits_{k=1}^\infty\frac{1}{X_1+\dots+X_k}$ with $(X_k)$ i.i.d. non integrable

Pick a sequence $X_1$, $X_2$, $\dots$, of i.i.d. random variables taking values in positive integers with $\mathbb{P}(X_i=n)=\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$ for every positive integer ...
1
vote
2answers
54 views

Continuity of probability measures for a process

Let $(B_t)_t$ be a Brownian motion, then I am given a stopping time $\tau_s:=\min(\inf\{t \ge 0; B_t=a\}, \inf\{t \ge s; B_t=b\}; \inf \{t \ge 0;B_t=c\}),$ where $a<0<b<c.$ Now, I want to ...
1
vote
1answer
25 views

General formula for $P(a_1 \le X_1 \le b_1,\dots, a_k \le X_k \le b_k)$ in terms of the joint CDF

I know that : $$ P(a_1 \le X_1 \le b_1,a_2 \le X_2 \le b_2) = F_{X_1 X_2}(a_1,a_2) + F_{X_1 X_2}(b_1,b_2) - F_{X_1 X_2}(a_1,b_1) -F_{X_1 X_2}(a_2,b_2) $$ Where $F_{X_1 X_2}$ is the joint cumulative ...
2
votes
2answers
44 views

Lemma about the probability space P (From Grimmett and Stirzaker)

This is from Grimmett and Stirzaker, Chapter 1, page 6. The triple $(\Omega,\mathscr{F},P)$ is a probability space. It has some important properties. (a) $P(A^C)=1-P(A)$ (b) if ...
1
vote
0answers
19 views

Finding the marginal distributions of a binormal random variable

Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where, $\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m ...
0
votes
0answers
20 views

Why is the stochastic integral $\int_0^t \nabla u(B_s)\cdot dB_s $ a local martingale?

This is from Durrett's book Stochastic calculus: a practical introduction. I don't understand the last sentence in the picture. Could anyone help explain why the first term is a local martingale? ...
1
vote
0answers
21 views

Normal $(\frac{n-1}{n})$-percentile asymptotic to $(2\log n)^{1/2}$?

I am working from Durrett's Probability: Theory and Examples, and I have encountered the following question: Suppose that $X$ is normally distributed, and $b_n$ is defined by ...
2
votes
1answer
92 views

Brownian motion: hitting times for closed sets are stopping times (and more).

Let $(B_t)$ be a $d$-dimensional Brownian motion, and consider the filtrations $(\mathcal{F_t^B}) = \sigma(B_0,...,B_t)$ and $\mathcal{F_t} = \cap_{\epsilon > 0}{\mathcal{F_{t+\epsilon}^B}}$ (the ...
1
vote
2answers
31 views

Uncorrelated and X given $Y = 0$

Is the following true or false? Suppose that $X$ and $Y$ are two discrete random variables defined on the same probability space. If $E[X] = E[Y] = 0$ and $E[X | Y=y] = 0$ for all $y\in Y$, then $X$ ...
0
votes
0answers
31 views

Showing a sequence converges in probability

I'm studying for a test on Monday and am going through some supplementary problems. These problems do not come with solutions provided but I still think they are very good practice. Suppose sequence ...
1
vote
0answers
60 views

Probability that Consecutive Partial Sums of Normals are Positive

Let $\{X_i\}$ be i.i.d. standard normals and let define $S_n = \sum_{k = 1}^n X_k$ to be their partial sums. Find $\mathbb{P}(S_1,S_2,S_3 >0)$. What I've Tried: We can set up the integral in a ...
3
votes
1answer
33 views

Calculating Expectation of Exponential with Indicator Functions (Continuous Time Martingales)

Update: So it turns out that the question I was given is actually wrong! $X_t$ should be defined with $\{T>t\}$ not $\{T<t\}$. My question is the following. It looks really easy, and so I ...
3
votes
1answer
74 views

Probability of roots of equation are real

Let $A$, $B$, and $C$ be independent random variables, uniformly distributed over $[0,9], [0,3]$ and $[0,5]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are ...
0
votes
1answer
47 views

Sweet - Question

There are n sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow. Hannah takes a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the ...
0
votes
1answer
53 views

How is the Inverse function theorem used to prove that the formulae in this question are the same?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x ...
0
votes
1answer
59 views

$X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite ...
2
votes
1answer
36 views

Exercise 1.2.4 (From Grimmett and Stirzaker)

This is yet another problem, where I have run into trouble. Let ${F}$ be a ${\sigma}$-field of the subsets of $\Omega$ and suppose that $B\in{F}$. Show that $G=\{A\cap{B}:A\in{F}\}$ is a ...
0
votes
1answer
41 views

Compute $E(Y\mid X=m)$ [closed]

The random vector $(X,Y)$ has the following joint distribution $$P(X=m, Y=n) = {m\choose n} \frac{1}{2^m}\frac{m}{15}$$ where $m=1,\ldots,5$ and $n=0,\ldots,m$. Compute $E(Y\mid X=m)$
2
votes
0answers
53 views

Odds of winning a lottery

My question will be similar to the one asked here Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse? I understand that each lottery is an independent event so selecting any ...
0
votes
1answer
98 views

Probability of a set that has infinite Lebesgue measure

Forgive, for the title didn't know how to name this questions. Please change to something better. Let $B_1(n)$ denote a unit ball around $n\in \mathbb{Z}^{+}$. Suppose that for every $n$ there exists ...
0
votes
1answer
38 views

Where is my error in finding the edgeworth expansion of the binomial distribution?

Let $B_n$ be a standardized binomial distribution. To illustrate the Edgeworth Expansion I made a plot showing $f(x)=P(B_n \le x)-\Phi(x)$ and $g(x)=\frac{p-q}6 (x^2-1) \phi(x) \frac{1}{\sqrt{npq}}$ ...
3
votes
2answers
53 views

Show that if $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ then $X_n \overset{P}\to X$.

Let $(a_n)\subseteq \Bbb{R}$ be a sequence such that $a_n \to \infty$. Let $(X_n)$ be a sequence of random variables such that $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ fore some random variables $X$ ...
1
vote
2answers
67 views

Expected value, E(X), where does the “x” in the formula come from?

So if we have the formulas $$Eg(X) = \int \cdots \int g(x_1,\dots,x_k)f_X(x_1,\dots,x_k) \, dx_1\dots dx_k$$ or, for discrete, $$Eg(X) = \sum_{x\in M} g(x)f_X(x)$$ with $X:\Omega\to\mathbb{R}^k$ a ...
0
votes
0answers
25 views

Renewal theory distribution of residual lifetime in delayed case with delay distribution of $F_0$

Show that $H(t,x)= F_0(t+x)-F_0(t) +F_0*H(t,x)$ where $H(t,x)$ is the probability that at least one renewal epoch between $t$ and $t+x$.Here $F_0(x)=\frac{1}{\mu}\int_0^x(1-F(x))dx$ where F is the ...
1
vote
0answers
15 views

Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variable with $\varphi$ density given.

Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variables with $\varphi_{n}(x)$, density given. $$\varphi_n(x)=\frac{n}{\pi(1+n^2x^2)}, x\in ...
0
votes
0answers
22 views

Asymptotic variance for Markov chain

Let $\hat{\mu_t}(f)=\frac{1}{t} \sum_{i=1}^{t} f(X_i)$ is an estimator for a finite, irreducible Markov chain $\{X_t\}$ with its stationary distribution $\pi$. In addition, assume the estimator is ...
0
votes
1answer
22 views

Proving $P(\bigcap A_i)\geq\sum P(A_i)-(n-1)$

I need to prove $P(\bigcap A_i)\geq\sum P(A_i)-(n-1)$. I tried playing with $\sum P(A_i)\geq P(\bigcup A_i)\geq\sum P(A_i)-\sum_{1\leq i\lt j\leq n}P(A_i\cap A_j)$, but didn't get anywhere. A hint ...
0
votes
0answers
35 views

Writing an event as a disjoint union of events - $\bigcup A\cap B_i$ or $\bigcup A|B_i$

I am having trouble with two ways of writing an event as a disjoint union of events - as the union of $A\cap B_i$ or $A|B_i$. For example: Say we have a bag with $a$ blue balls, $b$ red balls and ...
1
vote
0answers
31 views

Limit of expectation of L2 norm unbounded implies function ill defined

Given a random process $A(t,x) := \lim_{N\rightarrow\infty} A^N(t,x)$ where $ x \in (0,1)$ and $t \geq 0$ and given that \begin{equation} \mathbb{E}\left[ \int_0^1 A^N(t,x)^2 dx \right] = tN ...