Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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1answer
30 views

Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
-2
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2answers
76 views

Find the probability of $ x_2/x_3 \leq a $ where $x_2,x_3$ are uniform i.i.d.

Let $x_1,x_2,...,x_n $ be independent and identically distributed, uniformly on $(0,1)$. How to find $P(x_2/x_3 \leq a)$?
0
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1answer
67 views

$\lim_{n\rightarrow\infty}X1_{\{X>n\}}=0 \space\space\space\text{a.s.}$

$$\lim_{n\rightarrow\infty}X1_{\{X>n\}}=0 \space\space\space\text{a.s.}$$ This is a claim I need to use in part of a proof that involves the dominated convergence theorem. Since it holds almost ...
0
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1answer
382 views

Find unknown value in probability density function

Suppose that a random variable Y has a probability density function given by f (y) = ky^3*e^(-y/2), y > 0, and 0, elsewhere. Find the value of k that makes f (y) a density function. I found that k=1 ...
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1answer
26 views

How does arg max work in this context?

I'm implementing some stuff for machine learning and I ran across this post detailing some information on Bayes Theorem I was looking for: ...
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0answers
19 views

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. [duplicate]

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. From here http://math.stackexchange.com/a/878635/140308 (proof attempt is there too)
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0answers
12 views

exponential bound on the number of possible clusters at $0$ in $\mathbb{Z}^d$

Let us say that $\mathbb{Z}^d$ is given the usual lattice structure as a graph. I want to know the number of connected induced subgraphs of size $k>0$ that include the vertex $0$. Call this ...
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0answers
10 views

Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a markov chain related to the notion from PDE theory? For instance, if the markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
6
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3answers
133 views

Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
0
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1answer
37 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
0
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1answer
21 views

How to find the dynamics of stochastic process?

We have $Y_t=e^{\int_0^t W_sds}$. How do I obtain the dynamics of $Y_t$ (i.e. $dY_t$)? It seems that we can't use Ito Lemma because $\int_0^t W_sds$ is not in the form $X_t = \int_0 ^t \sigma_s dW_s ...
0
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1answer
23 views

Dynamics of short rate in HJM

According to a simplified HJM framework, we have: Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion. Dynamics of forward rate: ...
0
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0answers
19 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
6
votes
1answer
111 views

Sum of average reciprocal of which random variable converges to a Cauchy distribution?

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
8
votes
1answer
272 views

What happens to a random walk when we increase the probabilities of going right?

Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all ...
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0answers
12 views

exercise 1.21 of chapter 1 of Revuz and Yor's

This is the exercise 1.21 of chapter 1 of Revuz and Yor's: Let $X=B^+$ or $|B|$ where $B$ is the standard linear BM, $p$ be a real number $>1$ and $q$ its conjugate number ($q^{-1}+p^{-1}=1$). ...
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1answer
32 views

Proving independence of variables with normal distribution [on hold]

Random variable $X$ is a variable with standard normal distribution. How to prove that $|X|$ and $\frac{X}{|X|}$ are independent? Thanks.
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0answers
10 views

Mean time spent in a state in a Continuous-time Markov Chain

I consider a continuous-time homogenous Markov chain: with discrete state $X$ taking values in $\mathcal{F}=\{1,\cdots,N\}$ with the transition rates satisfying: \begin{equation} \begin{cases} ...
0
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1answer
62 views

Probability two people born on 1 April

Find the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April 1 exceeds 1/2. My answer: Total of n people chosen at random ...
1
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1answer
15 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
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1answer
50 views

Are the product and ratio of two characteristic functions still characteristic functions?

Let $\bf X $ and $\bf Y$ be random vectors (that may be dependent). Let $\varphi_{\bf X}(\bf t)=E[e^{i\sum X_it_i}]$ be the characteristic function of the random vector $\bf X$. (1) Is $\varphi_{\bf ...
0
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1answer
22 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
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1answer
24 views

Sum of Independent RVs in $L^1$

My friend and I were studying for a preliminary exam in probability and came across the following problem. Suppose $X$ and $Y$ are independent random variables. Prove that if $X+Y\in L^1$ then $X$ ...
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1answer
258 views

Aldous criterion for tightness in $D[0,1]$

Does anyone know where I can find some useful information about the Aldous criterion for tightness in the space of all cadlag functions $D[0,1]$?
3
votes
1answer
56 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
2
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1answer
55 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
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1answer
23 views

$\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots, S_n)$ generated by partial sums

is true that $\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots,S_n)$ where $S_n=\sum_{i=1}^n X_i$ in general or I have to impose additional restrictions to the random variables (for instance, independence)? ...
1
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1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
1
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3answers
29 views

Is $E[Bin(X,p)]=E[X]p$?

We have given some random variable $X$ with mean $E[X]=:\mu$. Now we are interested in a random variable $Y \sim Bin(X,p)$. Is it true that $$E[Y]=\mu p?$$ What confuses me is that normally the ...
2
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2answers
70 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
4
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1answer
66 views

Why if we use independence and factorization, we cannot represent every joint distribution? (rigorous argument needed)

I was reading Koller's Probabilistic Graphical models book and it says something like this: Let $P(x_i) = \theta_i$. Define: $$P(x_1, \ldots , x_n) = \prod_{i=1}^n \theta_i$$ This ...
0
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1answer
38 views

Inequality on characteristic functions (probability theory)

Show that for every real characteristic function $\phi(t)$ we have $$1-\phi(2t) \le 4(1-\phi(t))$$ I am not sure where to begin. It seems I am missing some formula or theorem, or is it really that ...
0
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1answer
37 views

Help proving $Pr(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$ if $ P \models (X \perp Y | Z)$ and $\mathcal{X}=X \cup Y \cup Z$

I was trying to prove the following: if $X,Y,Z$ were three disjoint subsets of variables such that $\mathcal{X}=X \cup Y \cup Z$, Prove that $ P \models (X \perp Y \mid Z)$ if and only if we can ...
0
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1answer
84 views

Expected number of coin tosses until a run of $k$ successive heads occurs

Suppose each coin toss is independent, what is the expected number of coin tosses until a run of "k" successive heads occur? Tried finding a recursive expression to solve the problem but got ...
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1answer
19 views

The equivalent condition of the almost surely convergence [on hold]

$X_n\rightarrow X$ a.s. if and only if, given $\epsilon>0$ and $\delta>0$, there exists $n(\epsilon,\delta)$ such that $\mathbb{P}\{|X_n-X|\geqslant\epsilon \mbox{ for some } n\geqslant ...
0
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0answers
13 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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1answer
18 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
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1answer
42 views
+50

What does the decomposition, weak union and contraction rule mean for conditional probability and what are their proofs?

I was reading Koller's book on Probabilistic Graphical Models and was wondering what the decomposition, weak union and contraction properties of conditional probability mean. Decomposition: $$(X ...
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2answers
2k views

Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
1
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1answer
33 views

Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
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0answers
78 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
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1answer
19 views

Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
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2answers
138 views

How can an element not be a member of its own equivalence class?

I'm working my way through these notes on stochastic calculus: The following is taken from section 2.20: In discrete probability, equivalence classes are measurable. (Proof: for any ...
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0answers
57 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
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2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
1
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1answer
40 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
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1answer
170 views

Mathematician who talked about the probability of a “good” graph?

In my undergraduate years, one of my professors always talked about this one mathematician who was talking about "good" graphs and wondered about the existence of such a graph. Apparently this ...
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2answers
2k views

probability density of the maximum of samples from a uniform distribution

Suppose $$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$ and suppose $$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$ How would I find the probability density of $\hat\theta$? Thank you!
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0answers
20 views

Integration respect the Lévy measure

How I can compute numerically $$\int_{a}^{b}f(z)\nu(dz)$$ where $\nu$ is a Lévy measure and $f$ is a continuous function? Is it equivalent to $$\int_{a}^{b}f(z)d\nu(z)$$ Thanks
6
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1answer
76 views

Concentration of measure vs large deviation

When reading some probability publications I am always not sure why they call this or that inequality a 'concentration inequality' or 'large deviation inequality'. For me these (concentration of ...