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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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 ...
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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 ...
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1answer
18 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 ...
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1answer
78 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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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 ...
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3answers
62 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
42 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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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 ...
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1answer
109 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 ...
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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 ...
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1answer
33 views

Law of large numbers for nonnegative random variables [closed]

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 $. ...
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1answer
43 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
42 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 ...
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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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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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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
57 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 ...
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3answers
49 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\} ...
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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 ...
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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 ...
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0answers
105 views

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

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 ...
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2answers
41 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 ...
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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 ...
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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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22 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
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2answers
48 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
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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
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1answer
41 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 $$ ...
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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 ...
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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, .., ...
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12 views

Properties of the inverse stable subordinator

I just need a good reference and/or some more well-known results (which I don't know yet) for the following situation. Let $\alpha\in(0,1)$. If I have an $\alpha$-stable subordinator $\mathbf{X}$, ...
2
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1answer
61 views

$\sigma+$-field of a Brownian motion

For a standard Brownian motion define \begin{align}\mathcal{F}_{0+} &= \bigcap_{t>0} \mathcal{F_t},\\ \mathcal{F_t} &= \sigma(W_s, 0 \le s \le t)\end{align} Which of the following ...
2
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1answer
59 views

$\sup_{t \in [0, \infty)} \left|[(H^{(n)} - H) \cdot X, Y]_t \right| \overset{P}{\rightarrow} 0$

1. Notation We start with establishing some (standard, I think) notation. Let $(\Omega, \mathcal{A}, P)$ be a given probability space. For any filtration $\mathcal{G} = (\mathcal{G}_t)_{t \in ...
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1answer
21 views

Expected Value and Variance of a GBM Function

What is the the expected value of the process $Y = X^{3}$, where X satises the SDE $$ dXt = −X_tdt + σX_tdB_t $$ $(σ > 0)$ and $X_0 = 1$ I have two different answers: 1) I know that $X_t$ is a ...
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30 views

Transformation Stratonovich to Itô SDE (for BM on a surface)

The question arises from a section to Stochastic Differential Geometry in Rogers L.C.G., Williams D. Diffusion, Markov processes and martingales. Vol.2. Itô calculus. (31.22) Brownian motion on a ...
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38 views

Correct steps in rewriting expectation to a probability

My knowledge of measure theory and probability spaces is limited, so please keep it relatively simple. Let $\{X(t), ~ t \ge 0\}$ be a Markov process on the countable state space $\mathbb{N}_0$ with ...
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1answer
31 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
0
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1answer
32 views

Show that for smaller $n$ than $n = 125000$ holds: $\mathbb{P}(|Z_n - \frac{1}{2}| \geqslant 0,01) \leqslant 0,02$

I'm a first year math student and I am having trouble with this exercise: Let $S_n$ be the amount of times we get heads when throwing a coin $n$ times. Let $Z_n = \frac{S_n}{n}$. With the equality ...
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2answers
43 views

How should I interpret this notation?

I am reading some lecture notes and I'm not sure how to interpret this: $$ b_j(x)=p(x\mid s_j)=N(x;\mu,\sigma^2)$$ It is clear from the context that $N$ refers to normal distribution, but what exactly ...
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2answers
60 views

How big is the chance that a arbitrary man is taller than a arbitrary woman?

I'm a first year mathematics student, and I'm having trouble with computing the following: Assume that in a country the height $X$ of men is normally distributed, with $\mu_X = 180$ (the expected ...
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7 views

Is any distribution of interarrival times stochastically dominated by a geometric random variable? [on hold]

If I have a process for which I do not know how the interarrival times are distributed: Is there always a geometric random variable that stochastically dominates the interarrival times? I think not ...
1
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1answer
25 views

Relations between the topologies of $L^0$ and $L^1$ on finite measure space

Let $(\Omega, \mathcal F, P)$ be a probability space and denote by $L^0(\Omega, \mathcal F, P)$ the space of all random variables $X : \Omega \to \mathbb R$ (i.e. measurable functions between ...
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0answers
19 views

Moment generating function less than infinity

Let $(X_i)$ be i.i.d. real-valued random variables satisfying $$ \mathbb{E}(e^{tX_1})<\infty. (*) $$ Does this condition mean that the moment generating function exists? Am I right, that showing ...
2
votes
1answer
27 views

Convergence in distribution with finite mean

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with another one of the exercises our professor gave us. Let $X_n, n=1,2,\ldots,$ and $X$ be nonnegative ...
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21 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...