New answers tagged

1 vote

Dudley's inequality: Sending $\delta$ to $0$

To deduce $\mathbb{E}[\sup_{t, s\in T\colon d(t, s)<\delta}X_t - X_s]\to 0$ as $\delta\to 0$, I think separability for the process suffices. Recall that we say a process is separable if there ...
pbb's user avatar
  • 90
0 votes
Accepted

Finding average tries to get a positive result while knowing probability of getting a positive result in n tries

To correct an error in the comments: Here, $p_n$ denotes the probability that you have observed at least $1$ success in the first $n$ trials. Thus, $1-p_n$ is the probability that you have observed $...
lulu's user avatar
  • 70.5k
1 vote

Exercise on Girsanov's theorem

There is no difference with the case your drift is non-deterministic. Your calculation shows that $X$ is a $Q$-Brownian motion; since $Q$ and $P$ are equivalent and $Q(\{X_t > M\}) > 0$ it ...
Jose Avilez's user avatar
  • 12.8k
1 vote
Accepted

Derive the 0-1 law from a functional equation of a martingale limit

For $z>0$ and $0<x<1$ you have $$ \lim_{t\to+\infty}\Bbb E(e^{-tM(z)})=\lim_{t\to+\infty}\Bbb E(e^{-tzx^{2z-1}M(z)})=\lim_{t\to+\infty}\Bbb E(e^{-tz(1-x)^{2z-1}M(z)})=\Bbb P(M(z)=0). $$ ...
John Dawkins's user avatar
  • 25.8k
1 vote

Can Chernoff Bound Theorem be applied to functions of independent random variables

I think McDiarmid inequality is what you are looking for. https://en.wikipedia.org/wiki/McDiarmid%27s_inequality
Ibra's user avatar
  • 140
0 votes

Show the Mean Squared Error of the Sample Mean

Old question but I'm posting the gory detail below in case any google searcher needs it. You might start with the knowledge that MSE = bias^2 + variance and compute them separately. Call your sample ...
stalactite's user avatar
2 votes
Accepted

Simple solution to random walk

One way to solve this problem is to represent the possible courses of the match by paths through a directed graph. Each vertex of the graph represents a possible score $\ (h,a)\ $ that might have ...
lonza leggiera's user avatar
-1 votes

When people saying 'with high probability', are there two possible interpretations?

"With high probability" means with a probability limiting to 1; this is not a vague phrase. The phrase which is vague is "very close". In the example you gave, the idea that they ...
Logan Post's user avatar
2 votes

Simple solution to random walk

With the $X$-axis representing score of the home team and the $Y$-axis that of the away team. write down scores as $0-3\mid 1-3 \mid 2-3 \mid 3-3 \mid 4-3$ $0-2\mid 1-2 \mid 2-2 \mid 3-2 \mid 4-2$ $0-...
true blue anil's user avatar
1 vote
Accepted

Calculating Probability in Disjoint and Independent Events

No, it's not true that $\ P(A'\cap C'\,|\,B)=\frac{P(A'\cap\,C')}{P(B)}\ .$ By definition $$ P(A'\cap C'\,|B)=\frac{P((A'\cap C')\color{red}{\cap B})}{P(B)}\ ,\tag{1}\label{e1} $$ and since \begin{...
lonza leggiera's user avatar
1 vote

Simple question about expectation

If $X$ is discrete and $P(X>0)>0$ then there exists some $x_i>0$ such that $P(X=x_i)>0$. As $P(X\geq 0)=1$, $X$ cannot take on a negative value with a non-zero probability, because the sum ...
Julio Puerta's user avatar
  • 4,376
2 votes
Accepted

PDF of $Y=g(X)$ when $X\sim N(0,1)$. $g(X)$ is a piecewise function where each part is constant.

Your expression for the CDF of $Y$ is close, but should be $$F_{Y}\left(y\right)=\begin{cases} 0, & y<-1\\ 1/2, & -1\leq y<1\\ 1, & y\geq1 \end{cases}.$$ Since $Y$ is discrete, it ...
AOS's user avatar
  • 171
1 vote
Accepted

Expected value and average value problem

If $$1-\Pr\left\{ 0.64\leq X\leq0.66\right\}$$ $$=1-\Pr\left\{ \dfrac{0.64-\mu}{\sigma/\sqrt{n}}\leq\dfrac{X-\mu}{\sigma/\sqrt{n}}\leq\dfrac{0.70-\mu}{\sigma/\sqrt{n}}\right\},$$ then $$\Pr\left\{ 0....
AOS's user avatar
  • 171
1 vote

is $P(A|B) = P(A|B \cap C)+P(A|B \cap C^{c})$?

The law of total probability applied to events $A$ and $B$ is $$\Pr[A] = \Pr[A \mid B]\color{red}{\Pr[B]} + \Pr[A \mid \bar B]\color{red}{\Pr[\bar B]};$$ that is to say, we must weight the conditional ...
heropup's user avatar
  • 136k
1 vote
Accepted

Does $X_nY_n=\mathcal{o}_{p}(\beta_n)$ hold?

The conclusion is true even if $X_{n}$ converges in prpbability to $0$ which is a weaker assumption than yours. You can prove it by in two ways. Method 1 Fix $\epsilon>0$ and a $\delta>0$. As $...
Mr.Gandalf Sauron's user avatar
0 votes

How to bound a sum of absolute value of centered Randemacher variables

Let $\xi_i=|a_i-2p+1|$, then $\xi=2(1-p)$ with probability $1-p$ and $\xi=2p$ with probability $p$. Hence $\mathbb{E}(\xi_i)=2(1-p)\cdot(1-p)+2p\cdot p=(2p-1)^2+1=:\mu$ and $\xi_i$s are i.i.d. Now you ...
van der Wolf's user avatar
  • 2,327
3 votes

Is the function $t \mapsto \mathbb P \left (\left |X - Y \right | > t\ |\ (X + Y) > t \right )$ monotone on $(0, \infty)\ $?

It is well known that if $X$ and $Y$ are i.i.d. $N(0,1)$ then $X+Y$ and $X-Y$ are independent normal variables. So $g(t)=P(|X-Y|>t)$. Further, $Z=\frac {X-Y }{\sqrt 2}$ is $N(0,1)$ too. So $g(t)=P(|...
geetha290krm's user avatar
  • 37.1k
2 votes

Walds identity and difference of stopping times

Consider the following (it technically doesn't match your requirements because $N_2 \leq N_1$, but I hope that doesn't matter much): $X_1, X_2, \ldots$ is a an i.i.d. sequence of Bernoulli random ...
Brian Moehring's user avatar
2 votes
Accepted

Rearranging exponentially distributed random variables by size

There is an elegant proof using thinning/merging of Poisson processes (and exponential racing that stems from those concepts), but I am not sure if OP is aware of this concept. So, let me tackle this ...
Sangchul Lee's user avatar
2 votes

Rearranging exponentially distributed random variables by size

Let $X_1,\dotsc, X_n$ be i.i.d exponetial random variables with rate $1$ and let $X\sim \text{Exp}(1)$. Define the random variables $$ Y_{i} = X_{(i)} - X_{(i-1)} \quad (1\leq i \leq n) $$ where $X_{(...
Sri-Amirthan Theivendran's user avatar
1 vote

Putting $k$ balls in n urns, which are grouped into $r$ sets.

Denote $R=2r+1$ and $m = \frac{n}{R}$ and $B={R \choose r+1}$. Let's do inclusion-exclusion with the sets $$ B_A = \{\text{all urns in groups in A are empty}\}, \text{ for } A \subset [R] \text{ with ...
ploosu2's user avatar
  • 8,842
1 vote
Accepted

Hypothesis Test using Confidence Intervals

You need to form the interval from the sample mean, not the total number of heads: $\hat{p} = \frac{9207}{17950} \approx 0.51 \implies \hat{\sigma}^2=\hat{p}(1-\hat{p}) \approx 0.25 \implies \frac{\...
Annika's user avatar
  • 6,908
0 votes
Accepted

A Gaussian process and a Rademacher proecss are sub-Gaussian

Gaussian Process. Let $Z=(Z_1,\cdots ,Z_d), t=(t_1,\cdots ,t_d), s=(s_1,\cdots, s_d)$. By definition, $Z_i \sim N(0,1)$ are independent Gaussian variables (Jointly normal random variables that are ...
Kaira's user avatar
  • 1,451
0 votes

Radon-Nikodym Derivative of a Mixed Distribution

The probability measure of an $X$ with the CDF you illustrated can be written as $\mu_X(A) = \mu_D(A) + \mu_C(A)$ where the discrete part is written $\mu_D(A) = \frac{1}{2} I_{0 \in A}$ where $I_{0 \...
Guillaume F.'s user avatar
0 votes
Accepted

Sub-Gaussian $X_t$, prove $\mathbb{E}\left[\sup_{t\in T}X_t \right] \leq 2 \mathbb{E}\left[\sup_{\rho(t,s)\leq \delta}(X_t-X_s) \right]+J(\delta,T)$

For this problem, bounding the expectation of the supremum is crucial. The following inequality can help in this case. Proposition. Let $\{Z_i\}_{i=1}^{N}$ be $\sigma^2$-sub-Gaussian random variables. ...
Kaira's user avatar
  • 1,451
1 vote
Accepted

How to prove an adapted Feynman Kac Formula for $v_t + \frac{1}2 \sigma ^2 (t,y) v_{yy} + b(t,y) v_y - \delta(t,y) v + h(y) = 0$ using SDE techniques?

We see $$\begin{aligned}dv(t,Y_t)-\delta(t,Y_t)v(t,Y_t)dt&=-h(Y_t)dt+v_y(t,Y_t)\sigma(t,Y_t)dW_t\\ \implies d(v(t,Y_t)e^{-\int_0^t\delta(s,Y_s)ds})&=-h(Y_t)e^{-\int_0^t\delta(s,Y_s)ds}dt+v_y(t,...
Snoop's user avatar
  • 15.2k
2 votes

Rigorous Mathematical foundations of Machine Learning / Deep Learning / Neural Networks

Don't know if you will find this interesting but it might be worthwhile giving Kevin Murphy's book(s) on probabilistic machine learning a try. https://probml.github.io/pml-book/book1.html
Oluwatobi Adefami's user avatar
0 votes

Exercise 4.3.11 of Durrett's Probability: Theory and Examples

Suggestion: Let $\pi:=P(\lim_nZ_n/\mu^n=0)$. Show that $\pi=\varphi(\pi)$ and then appeal to Theorem 4.3.12.
John Dawkins's user avatar
  • 25.8k
0 votes

Limit for Brownian local time

I think I have an answer: can you please let me know if you spot any mistake? Here we can write for every $n\in \mathbb{N}$, $L^{(n)}(t)=\frac{L(nt)}{\sqrt{n}}$ and $L^{(n)}(t)\overset{(d)}{=}L{(t)}$, ...
Randomwandering's user avatar
1 vote

The sum of two i.i.d. random variables cannot be uniformly distributed

When $X,Y$ are iid, the distribution of $Z=X+Y$ can be one of discrete or continuous uniform distributions, studied in the following. For the continuous case, two cases are considered: uniformity on a ...
Amir's user avatar
  • 4,694
1 vote
Accepted

Question on part of a proof using Chebyshev's inequality

Let $Y_n=X_2+X_3+\dots+X_n$, then $\mathbb{E}(Y_n)=0$ and $Var(Y)=\mathbb{E}(Y_n^2)=O(x)$ indeed. Hence by Chebyshev's inequality $$ \mathbb{P}(|Y_n|>\epsilon)\le \frac{Var(Y_n)}{\epsilon^2} $$ If ...
van der Wolf's user avatar
  • 2,327
2 votes
Accepted

Conditional probability of independent events and its interpretation

Firstly, the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$ is true regardless of whether $A$ and $B$ are independent or not. If $A$ and $B$ are independent, then the LHS becomes $P(A)$ and so we obtain the ...
Red Five's user avatar
  • 1,507
1 vote

Show that $ \mathbb P(X \in U)=\sup\limits \left \{\mathbb P(X \in K): K \subseteq U \text { is compact} \right \}.$

Your proof looks nice! I like your $K_n$ sets. I was confused about the meaning of "proper open" subset of $\mathbb{R}$, but I see now it just means an open subset of $\mathbb{R}$ that is ...
Michael's user avatar
  • 24k
3 votes
Accepted

Randomly choose $n$ numbers between $0$ and $1$. Add the smallest number from each repetition until the total is greater than 1. How many repetitions?

I don't think you can find a closed form solution for any $n$, but for small $n$ the closed form solution can be found: Like that answer you referenced, we first need the probability that the sum of ...
Ben's user avatar
  • 674
6 votes

The sum of two i.i.d. random variables cannot be uniformly distributed

For simplicity consider a random variable $Z$ uniform on $(-1,1).$ Its characteristic function is $$E(e^{itZ})=\frac {\sin t}{t}=\prod_{n=1}\left(1-\frac{t^2}{n^2\pi^2}\right).$$ If $Z\sim X+Y$ with $...
Letac Gérard's user avatar
0 votes

The sum of two i.i.d. random variables cannot be uniformly distributed

Part (1) Analysis Given $X$ and $Y$ are i.d.d. random variables uniformly distributed on $[0,\frac{1}{2}]$, we denote their common CDF by $F(x)$. For a random variable $Z = X + Y$, we analyze its ...
Saradamani's user avatar
  • 1,604
1 vote
Accepted

Finding the limiting distribution of $T_{n}/S_{n}$ as n tends to infinity

First note that $S_{n}$ is the sum of $n^{2}$ many iid $Poi(\lambda/n)$ variates and hence has $Poi(n\lambda)$ distribution. Now see that by Chebycheff's inequality, $$P(|\frac{S_{n}}{\lambda n}-1|\...
Mr.Gandalf Sauron's user avatar
2 votes
Accepted

function of random variables - change of variables

$$ \begin{align} \int f(z)p_Z(z)\,dz&=\mathbb E[f(Z)]\\ &=\mathbb E[f(g(X,Y))]\\ &=\int f(g(x,y))p_{X,Y}(x,y)\,dx\,dy\\ &=\int\int f(g(x,y))p_X(x)p_Y(y)\,dx\,dy, \end{align}$$ where I ...
Will's user avatar
  • 6,927
3 votes
Accepted

If $X, Y$ independent, and $Y$ has same distribution as $Z$, are $X, Z$ independent?

This is not true. Take $X$ and $Y$ i.i.d. and $Z=X$.
Célio Augusto's user avatar
0 votes

The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like

Because $\sup_{0<s\le t}W_s(\omega)/h(s)$ is monotone in $t$, you have $$ \{\omega: \limsup_{t\to 0}W_t(\omega)/h(t)\le 1 \} = \cap_{\epsilon\in\Bbb Q_+}\cup_{\delta\in\Bbb Q_+}\cap_{s\in\Bbb Q\cap(...
John Dawkins's user avatar
  • 25.8k
1 vote

Multivariate discrete pdf

Building on @StubbornAtom's comment, you have to indeed use that for a random variable $Y$ that follows a negative binomial distribution with parameter $r \in \mathbb{N}_{+}$ (so excluding 0, unlike ...
minginator's user avatar
1 vote

Characterizing sets of measure zero of the cylindrical $\sigma$-algebra.

The answer to your first question is no. For example, the union of two sets of measure $0$ also has measure $0$ but is not likely to be an intersection of cylinders. I doubt that there is a simple ...
Robert Israel's user avatar
1 vote

Randomly choose $n$ numbers between $0$ and $1$. Add the smallest number from each repetition until the total is greater than 1. How many repetitions?

Not a full answer... Let $R(x)$ be the expected number of tries to get an accumulated value that exceeds $x$. Then we can write the integral equation $$R(x) = 1 + \int_0^x R(x-u) \,g(u)\, du = 1 + \...
leonbloy's user avatar
  • 63.5k
1 vote
Accepted

Equivalent definitions for Support of random variables

Proof of $G_X \subseteq S_X$ under the assumption that $p_d$ is continuous: Suppose $x \in G_X$. If $x \notin S_X$ then $x \notin U_X$, so there exists $\epsilon >0$ such that $P_X((x-\epsilon, x+\...
geetha290krm's user avatar
  • 37.1k
1 vote
Accepted

Why is a factor of 24 used in this Central Limit Theorem Question?

if $X \sim N(0, \frac1{24^2})$, then $24X\sim N(0, 24^2 \cdot \frac1{24^2})$. Recall that $Var(aX)=a^2 Var(X)$, hence $Var(24X)=24^2\cdot Var(X)$. Also scalar multiple of a normal distribution remains ...
Siong Thye Goh's user avatar
0 votes

Creating a martingale given

You are correct, the filtration by construction makes the process adapted. You also need to check the integrability condition of the definition. To check the martingale property, you need to use the ...
Oscar's user avatar
  • 866
1 vote
Accepted

In this case does convergence of marginal distribution imply joint convergence in distribution

Clearly no. Consider $(S_n)_{n\geq 1} = (Z_n)_{n \geq 1}$ where $Z_n \sim \mathcal{N}(0, 1)$, a standard normal. Let $Z$ also denote a generic standard normal random variable. Consider $f(x)=x$ and $g(...
pbb's user avatar
  • 90
0 votes

Upgrade dominance in distribution to almost sure in a new probability space

It sounds like you are trying to keep the same $(\Omega, \mathcal{F})$, so the functions $X_i:\Omega\rightarrow\mathbb{R}$ and $Z_i:\Omega\rightarrow\mathbb{R}$ are always the same, but just change $P:...
Michael's user avatar
  • 24k
2 votes

The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like

The event you have captured in more explicit terms is not $\{\limsup_{t\to 0}W_t/h(t)=1\}$, but rather $\{\limsup_{t\to 0}|W_t/h(t)-1|=0\}$. The placement of the absolute value is crucial!
John Dawkins's user avatar
  • 25.8k
4 votes

Proof of a martingale condition regarding martingale transform

$$ \begin{align} C_n(\mathbb E[X_n\vert\mathcal F_{n-1}]-X_{n-1})&=\mathbb E[C_n(X_n-X_{n-1})\vert\mathcal F_{n-1}]\\ &=\mathbb E[(C\bullet X)_n-(C\bullet X)_{n-1}\vert\mathcal F_{n-1}]\\ &...
Will's user avatar
  • 6,927

Top 50 recent answers are included