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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3
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68 views

Calculation of Conditional Expectation $\Bbb E[f(X)\mid Y]$

$\newcommand{\Cov}{\operatorname{Cov}}$and thank you for taking the time to read this. :) My question is about evaluating $\Bbb E[f(X) \mid Y]$ (a random variable in $Y$). There's plenty online (and ...
0
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1answer
88 views

Size of families: Birth death immigration

The context of this problem is as follows. Starting from a population size of zero, immigrants arrive according to a homogeneous Poisson process with rate $\theta$. Once they arrive, immigrants start ...
1
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0answers
19 views

Comparing infinite dimensional distributions

Given two infinite sequences of rvs $(X_{1},X_{2},...)$ and $(Y_{1},Y_{2},...)$, how can we show $(X_{1},X_{2},...)\stackrel{d}{=}(Y_{1},Y_{2},...)$? The way I heard is by comparing all their finite ...
-1
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1answer
30 views

Show countable additivity of a certain probability measure

Let $\mathcal{F}$ be the field consisting of the finite and the co-finite sets in an infinite and ${\bf{uncountable}}\;\Omega$, and define a probability measure $P$ on $\mathcal{F}$ by taking $P(A)$ ...
0
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1answer
45 views

Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: $d(AB,CD)\leq d(A,C)+d(B,D)$ Remenber that: $(X,d)$---> Metric Space $d:X\times X\rightarrow ...
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3answers
70 views

If $X$ is a random variable with distribution $\mu$, prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \, \mu(dx)$.

I'm trying to prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \,\mu(dx)$ if $X$ is a random variable defined on $(\Omega, \mathcal{F}, P)$. I understand how to prove ...
0
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0answers
28 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
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0answers
20 views

Density of product of random variables

I am trying to calculate the product of two random variables, one that is exponentially distributed and the other that is uniformly between $[1, 2]$. Consider the following approach. We first ...
3
votes
2answers
110 views

Limit superior of $\sum_{j=1}^n X_j$ with $\mathbf{P}[X_j = 1] = \mathbf{P}[X_j = -1] = 0.5$

This is Exercise 2.3.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Let $(X_n)_{n\in N}$ be an independent family of $\mathrm{Rad}_{1/2}$ random variables (i.e., ...
0
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0answers
43 views

Entry time and hitting time

Hi I have a question about entry time and hitting time. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_{t})_{t \in[0,\infty)}$ be a $\mathbb{R}$-valued stochastic process on $(\Omega, ...
1
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2answers
20 views

When does probability mass outside a sufficiently large ball is small?

Many times when I read books about statistics or probability theory, I encounter proofs which said: For any $\epsilon>0$ there is an $M\in(0,\infty)$ such that $\text{Pr}\{X\in ...
2
votes
1answer
74 views

Correlation of a vector generated and its one-period lag, both generated using AR(1) data

Suppose that $C_0$ is $100$ and $\{e_t\}_{t\geq 1}$ is a sequence of i.i.d. standard normal random variables. We generate $C_t=C_{t-1}+e_t$ for $t\geq 1$ and set $$ x_t=C_t^2-C^2_{t-1},\quad ...
1
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0answers
44 views

How to find mean and variance for probability problem with warranty?

I am in a probability theory class and I'm stumped on a problem: A warranty is written on a product worth \$10,000 so that the buyer is given \$8000 if it fails in the first year, \$6000 if it fails ...
1
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0answers
48 views

Proof of Wald's Identity, is this valid?

So Wald says that assuming that $T$ a stopping time and $X_i$ i.i.d. variables are $L^1$, that $E[S_{T}] = E[T]E[X_1]$ given that $S_n = \sum_{i=1}^n X_i$. Consider the following proof that is ...
3
votes
0answers
67 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
0
votes
0answers
15 views

Cauchy distribution derivation

So now I'm doing a different example with the Cauchy distribution by letting $Z=\tan(U)$ for $U$ distributed between $[−π/2,π/2]$. So then $P(\tan(U)≤a)=P(U≤\arctan(a))$, which is equivalent to the ...
1
vote
1answer
54 views

How to approach analyzing probability problems? (Specific question included)

I've recently become very interested by the concept of probability. After doing some studying, I believe I've become fairly familiar with the terms: probability, random variables, probability ...
0
votes
0answers
22 views

Kullback-Leiber divergence for two simple probability vectors

For any probability vectors $ p= (p_1,...,p_K) $ and $ q=(q_1,...,q_K) $ representing monotonically increasing functions $ x-1 $ and $ ln(x) $ respectively what is the KL divergence? $$ KL(p||q) = ...
2
votes
1answer
30 views

Doob Decomposition is $L^1$ bounded

Suppose $X_n$ is a martingale that is $L^p$ bounded for some $p > 1$. Then the problem asks to show that the Doob Decomposition of the submartingale $|X_n|^p = M_n + A_n$ where $M_n$ is a ...
0
votes
1answer
212 views

Conditional expectation w.r.t. random variable and w.r.t. $\sigma$-algebra, equivalence

Let $\Omega = \{ \omega_i \}$ be a countable set, and consider some probability space $(\omega, \mathcal F, P)$ with $p_i := P(\{ w_i \})$. Let $X : \Omega \to \mathbb R$ be a random variable, then ...
0
votes
1answer
26 views

Convergence in distribution of a normalized Poisson distributed random variables

Show using the central limit theorem that $\frac{X_n-n}{n^{1/2}}\rightarrow Z$ where $Z$ is standard normally distributed and $X_n$ is $Poisson(n)$ distributed.
1
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1answer
26 views

Law of large numbers and identically distributed variables

I have trouble understanding the need of the "identically distributed variables" hypothesis in laws of large number's type theorem. For example, here ...
0
votes
1answer
24 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
9
votes
3answers
322 views

Expected Value of R squared

Let $n$ be a fixed positive integer. Generate $n$ numbers $x_1, x_2, ..., x_n$ from the set $[0,1]$, with the probability distribution being the uniform one and the $x_i$ all being independent of each ...
0
votes
0answers
12 views

Two iid random variables [duplicate]

Prove that for every two independent, identically distributed real random variables $X$ and $Y$, we have \begin{align*} P(|X-Y| \le 2) \le 3P(|X-Y| \le 1). \end{align*}
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0answers
41 views

distribution of the maximum of independent poisson random variables.

Let $X_i$ $i=1,\dots,n$ be independent poisson random variables with $X_i \sim \text{Poisson}(\lambda_i)$ then we define $X = \max_i X_i$ how does $X$ distribute? Is easy to see that ...
1
vote
1answer
50 views

Meaning of sampling i.i.d rvs from (random) probability measure?

Quote from book: "Consider an arbitrary atomic probability measure $\Gamma$ on unit sphere. Let $(\sigma_{l})$ denote an i.i.d sample from $\Gamma$." I don't understand the second sentence. Does it ...
2
votes
1answer
19 views

How to show that increasing r.v. imply stochastic dominance?

How can one prove the following statement: If $X$ and $Y$ are random variables such that $X(\omega) \geqslant Y(\omega)$ for all $\omega$ then $\mathbb P(X>x) \geqslant \mathbb P(Y>x)$ ? I saw ...
0
votes
0answers
17 views

The conditions on marginals that guarantees a certain class of measures

$x$ and $y$ are $m\times n$ matrices. $a, B,C$ are $m\times 1$ matrices. $b, A$ are $n\times 1$ matrices. $$\sum a_i=1, 0\leq a_i\leq 1, \forall i$$ $$\sum b_j=1, 0\leq b_j\leq 1, \forall j$$ ...
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0answers
34 views

Discrete measures and discrete kernels

This question was also posted here. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
2
votes
1answer
140 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
-2
votes
1answer
85 views

Given $B \cup A = B$ and probability and set theory axioms, prove $\mathbb{P}(A) \leq \mathbb{P}(B)$.

I need to prove that $\mathbb{P}(A)$ is less than or equal to $\mathbb{P}(B)$ using only this three things: $B \cup A = B$ The three axioms of probability: a) $\mathbb{P}(A)$ is greater or equal to ...
0
votes
0answers
32 views

Proper definition of Kullback-Leibler divergence for densities.

Let $f$ and $g$ be densities on the same set $X$. The Kullback-Leibler divergence is expressed by this famous formula: $$ D(f||g) = \int_{X^{+}} f(x)\log\left(\frac{f(x)}{g(x)}\right)dx \textrm{,} $$ ...
0
votes
1answer
29 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...
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0answers
44 views

Set of all probability measures with finite support

Let $X$ be an uncountable set endowed with the discrete topology. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, and consider the subset $A$ of $\mathcal{P}(X)$ consisting ...
-1
votes
1answer
83 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
0
votes
2answers
39 views

An $m$ sided dice is rolled $n$ times what is the chance of getting an average of $\frac{m+1}{2}$?

If I roll a six sided dice twice there is a $1$ in $6$ chance that the results will sum up to $7$ (giving an average of $3.5$ per dice). And if I only roll it once it is not possible to get a $3.5$. ...
1
vote
1answer
55 views

Optional Stopping Theorem for Stochastic Processes with Constant Mean (and not a Martingale)

Let $(X_n)_{n\geq 1}$ be a martingale with respect to $(Y_n)_{n\geq 1}$, i.e., the martingale condition $$ \mathbb{E}[X_n|Y_1, \ldots, Y_{n-1}] = X_{n-1} $$ holds. From this condition, it follows ...
0
votes
1answer
83 views

Expected value of the product of i.i.d random variables

Assume we have random variables $$X_i \,\,\,\ \text{ i.i.d } \,\,\ i\in[1:n]$$ with expected value $$\mathbb{E}[X_i] = \frac{1}{2}$$ Now let us compute the following expected value of the product of ...
0
votes
2answers
37 views

Expected delay problem on expectation based on uniform distribution

At a traffic junction, the cycle of traffic light is 2 minutes of green and 3 minutes of red. What is the expected delay in the journey, if one arrives at the junction at a random time uniformly ...
0
votes
1answer
58 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
2
votes
1answer
81 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
3
votes
1answer
35 views

Does wikipedia state the definition of probability correctly?

In the wikipedia article on probability http://en.wikipedia.org/wiki/Probability it says: To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a ...
6
votes
0answers
251 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
-1
votes
1answer
29 views

if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$?

In handouts provided by a professor I read: if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$. It does not feel right to me. $X ...
0
votes
2answers
48 views

What is the meaning of the below probability equation?

Can someone explain the intuitive idea behind this probability equation (especially the part where the limit of epsilon downarrow zero notation).
2
votes
1answer
50 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
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0answers
46 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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vote
3answers
31 views

Is this a conditional probability or not?

Suppose that the telephone calls during one minute time follow a Poisson distribution with mean=4. If people can handle at most 6 calls per minute, what is the probability that the people will receive ...
1
vote
2answers
44 views

infinite sum of normal r.v. is still a normal r.v. when given $\sum \limits_{i=1}^\infty a_i^{2}$ is finite

If $X_1, X_2, ...$ are i.i.d.standard normal random variables and for real constants $a_1, a_2, ...$, given $\sum \limits_{i=1}^\infty a_i^{2} $ is finite, then $Y_n =\sum\limits_{i=1}^n a_iX_i$ ...