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
314 views

How Borel sets make $\sigma$-algebra in a topological space?

I am trying to wrap my head around random variables and can't prove the following questions: How, in a topological space $(X, \mathcal{T})$, the collection of all Borel sets, say $\mathfrak{B}$, ...
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1answer
92 views

independence between an event and a set of events

On a probability space, from Billingsley's Probability and Measure, a family of classes of events is said to be independent, if we arbitrarily choose an event from each class in the family, and ...
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1answer
297 views

A question about the expected number of games being played

refer to this question I wanna change the question, if the rule of the game is the same, what is the expected number played to end the game?
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1answer
1k views

How to show that these random variables are pairwise independent?

Given the arrays $C=[C_1,C_2,...,C_N]$ and $S=[S_1,S_2,...,S_N]$ of lengths $N$ with elements that are discrete iid uniform distributed with equal probability (p=1/2) of being $\pm$ 1 Consider the ...
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1answer
3k views

Product and division of gamma distributions

SHORT VERSION: how would I go about solving the following inverse problem: $F(a,a_v,b,b_v,c,c_v)=\frac{A B}{C}$ where $A,B,C$ are gamma distributed according to A's mean=$a$, A's variance=$a_v$, etc. ...
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1answer
262 views

Transforming an inhomogeneous Markov chain to a homogeneous one

I fail to understand Cinlar's transformation of an inhomogeneous Markov chain to a homogeneous one. It appears to me that $\hat{P}$ is not fully specified. Generally speaking, given a $\sigma$-algebra ...
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1answer
659 views

Does this condition imply the Lindeberg condition?

Given a double array of random variables $X_{nj}, j=1,\dots, k_n, n\in\mathbb{N}$ with $k_n \to \infty$ as $n \to \infty$, suppose for each $n$, $X_{nj}, j=1,\dots, k_n$ are independent, each ...
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2answers
119 views

Compactness in $L^1$

Let $\mu(\cdot)$ be a probability measure in $X$. Consider a function $f: Z \times X \rightarrow \mathbb{R}_{\geq 0}$, with $Z \subset \mathbb{R}^n$ compact, such that: $\forall x \in X, \quad z ...
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2answers
182 views

Precise definition of random variables and probability measures

Suppose we have the probability space $(\Omega,\mathcal{A},P)$. Which of the following are right? $P$ is the probability measure defined on the events $\mathcal{A}$ as follows: ...
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1answer
511 views

Conditions for the disintegration theorem to hold

The disintegration theorem says that under certain conditions, a probability measure $\mu$ on a measurable space $$ the existence of Let $Y$ and $X$ be two Radon spaces (i.e. separable metric ...
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1answer
1k views

Law of Large Numbers and Cauchy Distribution

Let $\{X_n\}_{n=1}^\infty$ be a sequence of iid random variables under the Cauchy distribution with location parameter $0$ and scaling parameter $1$ (so that the density function is $f(x) = ...
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1answer
603 views

Conditional Expectation a Decreasing Function Implies Covariance is nonpositive

This comes from the book "A Probability Path". I'm just working through the problems trying to get a grasp of conditional expectations. Suppose $X,Y$ are random variables with finite second moments ...
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1answer
298 views

Necessary and Sufficient Conditions for Random Variables

I've been reading 2 textbooks in parallel on Probability Theory and they have 2 separate definitions of random variables $$ f:(\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{B}) \iff ...
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1answer
231 views

Elementary and measure definitions of conditional expectation and probability

In elementary probability, $E(Y \mid X =x)$ is defined as expectation of $Y$ w.r.t. the p.m. $P(A \mid X =x): = \frac{P(A \cap \{X=x \})}{P( X=x)}$ when $P( X=x) \neq 0$. ...
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1answer
268 views

Question for understanding definition of point process

I am trying to understand the definition of point process when reading its Wikipedia article: Let $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra ...
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3answers
1k views

Almost sure convergence

{Xn} is a sequence of independent random variables each with the same Sample Space {0,1} and Probability {1-1/$n^2$ ,1/$n^2$} Does this sequence converge with probability one (Almost Sure) to ...
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55 views

Does $P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3\right)$?

This link shows that that $P(X_i = x) = 0$ so can we say, \begin{equation} P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3\right) \end{equation} Assumptions $X_i, X_j$ are random ...
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0answers
17 views

If $F_+^{-1}$ and $F_-^{-1}$ are the right- and left-cont. inverse of a CDF, resp., then $\left\{F_-^{-1}(y)<X\le F_+^{-1}(y)\right\}$ has prob. $0$

Let $X$ be a real-valued random variable and $$F(x):=\operatorname P[X\le x]\;\;\;\text{for }x\in\mathbb R\;.$$ Moreover, let $$F_-^{-1}(y):=\sup\left\{F\le ...
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1answer
52 views

Asymmetric Random Walk / Prove $E[T] = \frac{b}{p-q}$ / How do I use hint?

Given random variables $Y_1, Y_2, \ldots \stackrel{\mathrm{iid}}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr ...
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1answer
38 views

Asymmetric Random Walk / Prove $E[X_{T \wedge n}] = (p-q)E[T \wedge n]$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
46 views

Symmetric Random Walk / Find $E[X_S]$ and $E[X_T]$

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = ...
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2answers
69 views

Does $\sup_{n \ge m} Y_n < \infty \ \text{a.s.}$ imply $\sup_{n \ge 1} Y_n < \infty \ \text{a.s. ?}$

Suppose $Y_1, Y_2, ...$ are independent random variables and $\exists m \ge 1$ s.t. $$\sup_{n \ge m} Y_n < \infty \ \text{a.s.}$$ Does this mean $$\sup_{n \ge 1} Y_n < \infty \ ...
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1answer
62 views

Show that if $X \sim N(0,1)$ and $Y \sim N(0,1)$ then $\frac{X}{Y} \sim \mathcal{C}(0,1)$ [duplicate]

Let $X,Y$ be independent random variables. Show, without using the change of variable theorem, that if $X \sim N(0,1)$ and $Y \sim N(0,1)$ then $\frac{X}{Y} \sim \mathcal{C}(0,1)$ So this problem is ...
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1answer
44 views

Probability of a independent sequence of random variables [closed]

Let $(X_n:n=1,2,\ldots)$ be an independent sequence of random variables, where, for each $n$, $X_n$ is uniformly distributed on $[0,n]$. Calculate $P(\{\omega:X_n(\omega)\to \infty \text{ as } ...
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1answer
39 views

Proving almost sure convergence

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ ...
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1answer
91 views

Probability of a set that has infinite Lebesgue measure

Forgive, for the title didn't know how to name this questions. Please change to something better. Let $B_1(n)$ denote a unit ball around $n\in \mathbb{Z}^{+}$. Suppose that for every $n$ there exists ...
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1answer
33 views

Where is my error in finding the edgeworth expansion of the binomial distribution?

Let $B_n$ be a standardized binomial distribution. To illustrate the Edgeworth Expansion I made a plot showing $f(x)=P(B_n \le x)-\Phi(x)$ and $g(x)=\frac{p-q}6 (x^2-1) \phi(x) \frac{1}{\sqrt{npq}}$ ...
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1answer
50 views

Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$, let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, ...
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1answer
71 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
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1answer
59 views

Find pdf of sum of n indp exp RVs w/o using MGFs

From Williams' Probability w/ Martingales: Re $E[f(S_n)]$, how do I obtain $f_{S_n}(s)$? It seems that $$f_{S_n}(s) = \frac{s^{n-1} e^{-\lambda s} \lambda^n}{(n-1)!}$$ I tried computing ...
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1answer
44 views

What is the relevance of linearity and monotone convergence theorem?

From Williams' Probability w/ Martingales: 5.5 is linearity of integration for non-negative $\Sigma$-measurable functions, and MON is monotone convergence theorem. What are the relevance of ...
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1answer
30 views

Determine measurability of E(X|N) or even $\sigma(E(X|N))$?

Suppose $(\Omega, F, P)$ is a sample space, $X$ a random variable, and $N$ a sub sigma algebra of $F$. How can we determine $\sigma(E(X|N))$? How is $\sigma(E(X|N))$ related to $\sigma(X)$ and ...
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1answer
130 views

Does $X ⊥ Y \leftrightarrow X ⊥ Y | Z$ implies $(X,Y) ⊥ Z$?

Let $X, Y$ and $Z$ be random variables. Let $p_1$ be the statement that $(X,Y) ⊥ Z$ (meaning $(X,Y)$ and $Z$ are independent), $p_2$ be the statement that $X ⊥ Y$ (meaning $X$ and $Y$ are ...
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1answer
58 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuous stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
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1answer
52 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
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1answer
47 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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1answer
390 views

Expectation of a discrete random variable: how to convert an integral to a sum?

According to wikipedia and all my textbooks, we define the expectation of a random variable on a probability space $(\Omega, \mathcal{F},P)$ : \begin{align} E(X) &= \int_{\Omega}XdP\\ \end{align} ...
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1answer
54 views

Does convergence in distribution of discrete random variables with same finite support imply convergence in probability?

Let ${X_m}\mathop \to \limits^D Y$ and ${\text{supp}}\left( {{X_m}} \right) = {\text{supp}}\left( Y \right) = \left\{ {0,1, \ldots ,n} \right\},\forall m \in \mathbb{N}$. Does ${X_m}\mathop \to ...
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1answer
43 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
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1answer
377 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
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1answer
293 views

Covariance of states of a finite Markov chain

I know it is possible to construct a covariance matrix for states of a Markov chain but I cannot seem to find a proper way to compute it. I will attach some theories I found from Kemeny and Snell's ...
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2answers
44 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
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2answers
95 views

If a probability space has no measurable subsets with $P$ strictly between $0$ and $1$, then every random variable is constant a.s.

Let $(\Omega, \mathfrak{F}, P)$ be a probability space such that $\forall F \in \mathfrak{F}, P(F) = 0 \ or \ 1$. Show that for all random variables X on $(\Omega, \mathfrak{F}, P)$, $\exists \ c ...
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1answer
152 views

An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
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44 views

What is the distribution of $Y_n$ and its convergency

Let $X_n$ be a iid sequence of Poisson random variables with parameter 1. Define $Y_0 = 1$ and $Y_n := X_nY_{n-1}$ for $n\geq 1$. How to show that $Y_n$ converges to $0$ almost surely, please? I think ...
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2answers
376 views

Conditional expectation of indicator function

Could someone confirm if the following is correct. If not why? \begin{equation} E[\mathbb{1_{X\leq x}}|Y]=P[X|Y]=\frac{P[X,Y]}{P[Y]} \end{equation} Thank you.
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2answers
118 views

A shop sells three types of light bulbs. A customer get a random light bulb - what is the mean-lifetime and variance of the light bulb?

Exercise: A shop sells three types of light bulbs. The lifetime for the three light bulbs is exponential distributed with mean $1, 1.2$ and $1.4$, respectively. In the store $25$ % of the light bulbs ...
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1answer
91 views

Conditional expectation of a stochastic process in filtered space

It was suggested* to me that if we have a stochastic process with independent increments, and $T > t$, then $$ E(X_{T-t}| \mathcal{F}_t) = X_{T-t} $$ where $\mathcal{F}_t$ is the filtration at time ...
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0answers
76 views

The subspace which $X-E(X|\mathcal G)$ is orthogonal to, and the set of r.v.s generating the same $\mathcal G$.

$X$ is a random variable. I wonder if there are some or no relations between the subspace which $X - E(X|\mathcal G)$ is orthogonal to, which is the set of all random variables which are both ...
0
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2answers
315 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...