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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1answer
33 views

Probability inequality problem about discrete random variable

Here is the problem. Let X be a discrete random with $\ E(X) = 0$ and $\ \text{Var}(X) = \sigma^2 < \infty $. Show that $$ P(X \geq a) \leq \frac{s^2}{s^2 + a^2} $$ for all $\ a > ...
1
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0answers
27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
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0answers
16 views

correlated random vectors converge to an independent random vector

Suppose $(X_k,Y_k)_{k\geq 1}$ is a sequence of random vectors. $X_k,Y_k$ have support $[0,1]$ and a joint density function $ f_k(x,y)$ being positive for all $x,y\in [0,1]$ and continuous everywhere. ...
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1answer
41 views

Expected number of dice throws to fill out a table.

Say I have a table of numbers ${1,2,3,4,5,6}$. Every time I throw a fair die, if the position in the table is unchecked, it becomes checked and if it is already checked it becomes unchecked. For how ...
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0answers
21 views

Expected value of signals

I am trying to learn DPS. A couple of explanations are based on statistics. I would like to understand what is and how coherence works, but I am stuck on its definition. I found the following ...
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2answers
59 views

Random equation-does it make sense?

What is the probability that the equation $$x^2+2bx+c=0$$ has real roots? Answer is exactly $1$. (or $100$%) For example: if $b=1$ and $c=2$ roots are complex. Does it make sense? If $P(A)=0$, then ...
-1
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0answers
29 views

Prove continuous stopped process $X_{T\wedge t}$ is a martingale if $X_t$ is a martingale [closed]

Looking for help proving that a continuous stopped process $X_{T\wedge t}$ is a martingale if the underlying process is a martingale. Any help is appreciated!
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1answer
26 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n ...
1
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1answer
28 views

Expected time to fill a table [duplicate]

Say I have a table of numbers 1-6. I throw a 6 sided die a number of times. Each time I get a number I have not already had, I mark it in the table. What is the expected number of times to throw the ...
2
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0answers
34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
2
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2answers
36 views

Var$(X) = \mathbb{E}((X - \mathbb{E}(X))^2) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$

I have a question about something my teacher told us: let $\mathbb{E}$(X) donate the expected value of a certain random variable $X$. Then Var$(X) = \mathbb{E}[(X - \mathbb{E}(X))^2] = ...
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4answers
43 views

Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$.

I'm having trouble with solving this problem: Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$. I know ...
0
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2answers
31 views

Take $k$ shoes ($k \leqslant n$) from a wardrobe. What is the expected value of the number of pairs ($X$) you take?

I'm having trouble with this question: Let there be $2n$ shoes ($n$ pairs) in a wardrobe, arbitrary ordened. Take $k$ shoes ($k \leqslant n$) from that wardrobe. What is the expected value of the ...
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0answers
27 views

$\lim_{n\rightarrow\infty}\operatorname{E}(X_n^2)=0$ implies $\lim_{n\rightarrow\infty}\operatorname{E}(X_n)=0$

Why $\lim_{n\rightarrow\infty}\operatorname{E}(X_n^2)=0$ implies $\lim_{n\rightarrow\infty}\operatorname{E}(X_n)=0$? Is it really that simple?
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1answer
51 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
1
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1answer
20 views

Determining the population size of a branching process

Suppose that I have the following branching process. Each parent can have up to two children, the number of which is determined by two independent fair coin flips. I know that this branching process ...
-3
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0answers
14 views

How to Randomly Generate SAT Scores in R with max and min? [migrated]

I'm trying to figure out how to randomly generate SAT scores in R by subsection. It follows the general form rnorm(n,mean,SD), but I also need to take into account that the minimum value has to be 200 ...
0
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1answer
17 views

In throwing 6n dice, what is the probability of getting each face n times? Use Stirling's formula to estimate this probability for a large n.

This question is taken from Probability Theory: A Concise Course by Y. A. Rozanov. My attempt at a solution is as a following: I think of $6n$ dice rolls as $n$ groups of 6 rolls. The probability ...
3
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1answer
76 views

Uniform distribution on a sphere

Consider the unit ball $S_n$ (centered at the origin) in $\mathbb{R}^n$ for $n \ge 1$ and a stochastic process $(X_t)_{t\ge 0}$ taking values in $\mathbb{R}^n$. Let $T = \inf\{t > 0 \colon X_t ...
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0answers
38 views

Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
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1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
0
votes
3answers
60 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity ...
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2answers
41 views

time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
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0answers
14 views

Continuous map of cadlag functions (one sided limits exist and right continuous) is cadlag

Recall that a real function $f$ is cadlag if the one sided limits $f (t^-), f (t^+)$ exist and $f (t^+) = f (t)$, i.e. $f$ is right continuous. Then is the following true? If $f$ is continuous and ...
5
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1answer
105 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
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0answers
22 views

The length of the set which will be covered by Brownian motion in a time $t$

I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable ...
0
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1answer
33 views

Law of large numbers for nonnegative random variables [on hold]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
1
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1answer
42 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
3
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1answer
41 views

Why is the supremum a random variable in the Glivenko–Cantelli theorem

According to wikipedia: Assume that $X_1,X_2,\dots$ are independent and identically-distributed random variables in $\mathbb{R}$ with common cumulative distribution function $F(x)$. The empirical ...
0
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1answer
62 views

mutual information adds along path

Is it true that $I(X;Y)+I(Y;Z)=I(X;Z)$ for $X \to Y \to Z$? $I(X;Z) = H(X)+H(Z)-H(X,Z)$ and $I(X;Y)+I(Y;Z) = H(X)+H(Z)-H(Z|Y)-H(X|Y)$ Hence, we would require $-H(X,Z)=-H(Z|Y)-H(X|Y)$ -- is it true? ...
2
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3answers
57 views

Computing $p(d|e_1,e_2)$ from $p(d|e_1)$ and $p(d|e_2)$

I know the probability $p(d|e_1)$ and $p(d|e_2)$, how to compute the $p(d|e_1, e_2)$ if $e_1$ and $e_2$ are independent? What if $e_1$ and $e_2$ is dependent, how to compute?
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0answers
34 views

Is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$ for continuous $X$

For continuous $X$ with distribution $F$, is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$? Also is continuity required? I've attempted a proof: Since $\mathbf{P} (F ...
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0answers
13 views

Are variables in embedded space Statistically independent variables?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
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1answer
56 views

Constructing a new Markov chain from another Markov chain

I have a very simple problem, but it seems I have difficulty to prove it rigorously. Suppose random variables $X, Y$ and $Z$ form the following Markov chain: $X\leftrightarrow Y\leftrightarrow Z$. My ...
3
votes
3answers
48 views

A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.

I'm having some trouble following someone's proof of the following result: Assume that $\{X_n\}$ are uniformly integrable and that $X_n\to X$ a.s.; then $EX_n\to EX$. First, the author shows that ...
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1answer
23 views

A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
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0answers
46 views

Why would one not want their probability space's sigma algebra to be the power set?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. It seems like we would want to be able to measure the probability of each $\omega \in \Omega$, for which we must require that each $\{\omega\} ...
0
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1answer
25 views

From pairwise P(A > B), to P(A > all distributions in set)

$\{D_0,…,D_n\}$ is a finite collection of independent (but not necessarily identically distributed) random variables. Define $f(x,y)=P(D_x≥D_y)$ and $g(x)=P(∀y:D_x≥D_y)$. Does $f$ determine $g$, and ...
0
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1answer
27 views

How many loops? Expected value

I have a problem with this exercise. I completely do not know hot to tackle it. Please help. A bin contains $N$ strings. You randomly choose two loose ends and tie them up. You continue until there ...
2
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1answer
36 views

Show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s.

Given a sequence $(X_n)_{n\geq 1}$, show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s. Here is my attempt: $|X_n|\leq Y$ a.s. means that $P(|X_n|>Y)=0$, $\forall n\geq 1$ $P(\sup_n ...
3
votes
0answers
83 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure [migrated]

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
1
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2answers
40 views

Why second moment about mean is better at describing spread than the first one?

Dispersion is usually used as a measure of inaccuracy of a measurement. It's defined as second moment about mean. Why not define dispersion as cubic root of third moment about mean or as first moment ...
1
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1answer
28 views

One Martingale problem

In the setting of Kolmogorov's maximal inequality, I need to prove the following $$P(\max_{1\leq m \leq n}|S_m| \leq x) \leq \frac{(x+K)^2}{var(S_n)}$$ Hint: Use the fact that $S_n^2 -s_n^2$ is a ...
1
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0answers
32 views

Showing that a characteristic function is “positive semi-definite”

I found this exercise while preparing for an exam. Let $X$ a real random variable, let $\varphi(t):=E[e^{itX}]$ its characteristic function. Show that $$\forall ...
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0answers
21 views

Can we do better than zero padding of FFT?

My background is in signal processing and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
2
votes
2answers
47 views

Mean Square Error of Monte Carlo

Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself a bit lost when going through a simple proof in the literature. I am working in the context of ...
2
votes
2answers
297 views

Self-independent random variable

Let $X$ be a self-independent random variable. Show that $X$ is almost sure constant. My proof (by condradiction): Assume that there are two disjoint Borel set $A$,$B$ such that: $\Pr(X \in A)>0$ ...
3
votes
1answer
40 views

Why do we know that $\left\{\lvert X_n-X\rvert >\epsilon\right\}$ is an event?

I hope my question is not too stupid. By definition a sequence $(X_n)$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon >0$ we have $$ ...
0
votes
0answers
27 views

Central Limit theorem: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected [duplicate]

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
0
votes
1answer
19 views

Are RV having same exp. value and covariance already have the same distribution?

Let $(X_1, ..., X_n), (Y_1, ... , Y_n)$ be random variables. $X_i$ has the same distribution as $Y_i$ for all $i$. $\forall i, j: Cov(X_i, X_j) = Cov(Y_i, Y_j)$ Do $(X_1, ..., X_n)$ and $(Y_1, .., ...