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 [tag:probability-...

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1answer
40 views

Is this function measurable with Q numbers?

Is this function measurable $$f : \mathbb{R} \rightarrow \mathbb{R}, x \mapsto \begin{cases} 0&x \in\mathbb{Q}\\ x&\text{ otherwise}\end{cases}$$ $\ss (\mathbb{R})-\ss (\mathbb{R})$$ - ...
0
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2answers
29 views

What's the probability a die irolled 4 times you will get only two kinds of faces?

Let $A$ be the event "only $2$ different faces in $4$ rolls of a die." At each roll there's $6$ possibilities, so: $$\omega = 6\cdot 6\cdot 6\cdot 6$$ Considering that it can be only two kinds of ...
0
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2answers
86 views

What is the probability that exactly 7 of the first 10 coin tosses are heads?

A biased coin is tossed infinitely many times and has probability $p$ of being "heads". What is the probability that exactly $7$ of the first $10$ coin tosses are "heads", in terms of $p$? It's a ...
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0answers
23 views

Link between zipfian distribution and probability of accessing a single object

I'm trying to understand Zipf's law in more detail, and more specifically, the relationship between: 1) a set of accesses to objects $\{A,B\}$ following a zipfian distribution over a number of ...
1
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1answer
166 views

Counterexample in optional stopping martingale

Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable. ...
2
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1answer
182 views

Coupon Collector Problem for Non-Uniform Coupons: On the number of missed Coupon

Suppose $\mathcal B=\{1,2,\ldots,b\}$ is the set of all possible coupons, with $\mathbf p = ( p_1,p_2,\ldots,p_b)$ assigning the probability of occurrence for all coupons in $\mathcal B$. The "...
2
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0answers
67 views

Pathwise integral of $W^{-a}$

Denote by $\tau(x) := \inf \{t \ge 0, W_t=x\},$ where $W_t$ is a Wiener process started at $W_0 = w_0 > 0$ and I would like to show that for any $a>1$ it almost surely holds that $$\int_{0}^{\...
0
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1answer
52 views

Are these random variables independent?

Just as in the central limit theorem, we let $X_i$ be iid with mean zero and variance $\sigma^2$, $S_n=X_1+\cdots +X_n$ and let $Y$ be the limit in distribution of $S_n/\sigma\sqrt n$. I think that $...
1
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0answers
81 views

Size-Biased Galton Watson Tree.

First of all, i am not sure whether this question belongs here or to stack overflow. Let me write here, first, i will give a definition of size-biased distribution. then i will give definition of size-...
1
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1answer
51 views

Transience in a simple Markov chain

Consider the following simple game from a textbook called "Competitive Markov Processes" by Filar & Vrieze (Springer 1996). This is a two player game with two states. In the first state (the ...
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2answers
92 views

If a Markov chains converges then the limit is a stationary distribution

Let $p$ be a transition function of a Markov Chain on a countable state $S$ and $i \in S$. Assume for every $j \in S$, $$ \lim_{n\to \infty} p^n(i,j) = \pi(j)$$ Show that $\pi$ is a stationary ...
2
votes
1answer
80 views

Conditional expectation w.r.t measure and push-forward measure

I have been introduce to the theory of conditional expectation. My question is simple. I wish to know the similarities between two quantities. My book talks about the existence of $E(Y|X=x)$ and $E(Y|\...
1
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1answer
219 views

Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$

The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$ And I know how to calculate the first total variation of the standard ...
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2answers
82 views

Does convergence of means imply convergence in mean?

For a sequence of nonnegative integrable random variables, I know that convergence in mean (aka convergence in $L^{1}$ where $E[|X_{n} - X|]$ approaches zero as n gets large), implies convergence of ...
1
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0answers
38 views

Does this converge in probability?

I'm asked whether $X_{1}, X_{2}, ...$ converges in probability where $P\left(X_{n}=\frac{1}{n}\right)=1-\frac{1}{n^{2}}$ and $P\left(X_{n}=n\right)=\frac{1}{n^{2}}$. I think I have the solution, but ...
1
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2answers
70 views

Show that $\limsup|Y_{1}+…+Y_{n}|/n = \infty$ almost surely

Can someone help me with part c) of question 2.8 located here (a 2005 probability course from Warwick University): https://homepages.warwick.ac.uk/~masgav/teaching/pm05_sheet2.pdf The question is: ...
1
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1answer
146 views

What is the joint distribution of two random variables?

Today I was thinking about this and I have the feeling I am missing something obvious, but I can't seem to solve it. Suppose we have a continuos random variable $X$ with density $f_X(x)$. Let $Y = g(...
1
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0answers
55 views

Geometric probability with square

Jhon and Simon have common bank account which has $720$ dollars. Each of them has to buy a gift independently from other (Gift cost $< 360$ dol). What is the probability that after shopping there ...
1
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1answer
116 views

The expectation of an infinite series of random variables [closed]

Let $(X_n)_{n = 1}^\infty$ be a sequence of random variables taking values in the extended real line, and suppose that $\sum_{n = 1}^\infty |E(X_n)| < \infty$. (a) Is it possible that $\sum_{n = 1}...
-2
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1answer
28 views

Probability of a gaussian distrbution

Assume that we have two variables $V_0$ and $\Delta V$, we know that $V_o$ follows a gaussian probability distribution. So what would be the $P(V_o< \Delta V)$ knowing $\sigma_o$ (The standard ...
2
votes
1answer
112 views

Uniformly distributed points on spherical surface

Let $x=(x_1,\ldots,x_n)$ be uniformly distributed on the $(n-1)$-dimensional spherical surface $S^{n-1}(n^\frac{1}{2})$ of radius $n^\frac{1}{2}$. I'm trying to show that as $n\to\infty$, $x_1$ ...
2
votes
1answer
81 views

Brownian bridge with multiple possible end values

Brownian bridge $Z_t$ is a diffusion process distributed as Brownian motion $B_t$ conditioned on the event $B_1 = 0$. It is rather well-studied, and allows for a Markov-like SDE representation. I ...
1
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1answer
204 views

Expectation and Variance of random walks

Consider random walks of fixed length (e.g. $5$) starting at node $u$ in an undirected and connected graph with $N$ vertices. If a node $k$ has $N_k$ edges, the probability of the walk reaching any of ...
5
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0answers
63 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
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0answers
45 views

Universal localization argument for polynomially bounded functions of a stochastic proceess

As in many mathematical disciplines, many statements about stochastic processes are really easy to prove for bounded objects (sets, function, processes, etc.). This is why many proofs (e.g. for Ito's ...
2
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0answers
39 views

Find the distribution of the random variable given probability generating function

Let $X$ be a non-negative, integer valued random variable such that $\phi_X(t)=-\log(1-qt)$. Determine $P(X=k)$ where $k=0,1,2,...$. Now nothing is said about $q$, so maybe we can assume $q$ such ...
0
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0answers
158 views

Kurtosis of Poisson

I'm trying to show that the kurtosis if X is $3+\lambda^{-1}$ for a Poisson. I think I should start by calculating the fourth moment around the mean and then at some point rewrite it in terms of ...
2
votes
1answer
34 views

Expected value is affine in distribution

How to show that $E_{P_X}[X]$ is affine in $P_X$. That is for two distributions $P_{X,1}$ and $P_{X,2}$ we have that \begin{align*} E_{\alpha P_{X,1}+(1-\alpha) P_{X,2}}[X]=\alpha E_{ P_{X,1}}[X]+(1-\...
0
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1answer
39 views

Probability that a joyride is overbooked

A bus operator knows that for every person the probability that a place is reserved, but not taken up, is 5%. The entrepreneur sold for a joyride 320 tickets, but in reality there are only 300 seats. ...
2
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0answers
49 views

Covariance of minimum of independent random variables and a constant

I have two random variables $X \sim \min (k, X_1)$ and $Y \sim \min (k, X_2)$ where $X_1$ and $X_2$ are exponential random variables with same rate $\lambda$ independent of each other and $k$ is a ...
1
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2answers
59 views

Is the following statement true about probabilities and their complements?

I saw the following statement written, but I can't understand why it is true. $$ \dfrac {P(A \text{ and } B)}{P(B)} = \dfrac{P(A)-P(A \text{ and }B^c)}{ 1-P(B^c)} $$ Any help understanding why these ...
2
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0answers
38 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} g(...
0
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0answers
44 views

Is this function a joint distribution function for two random variables? [duplicate]

Is the following function $F(x,y) = 1-e^{-xy} $ for $x,y \geq 0$ $F(x,y) = 0 $ otherwise a joint distribution function for two random variables $X$ and $Y$? Why?
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0answers
61 views

binomial sum bound

Let $X$ be a binomial random variable with parameters $n$ and $p$ that is $X \sim \mathrm{bin}(n,p)$. Does anyone know of a good lower bound (interms of $n$ and $p$) on the probability that $P(X \geq ...
2
votes
0answers
113 views

a.s. for all $t$ or for all $t$ a.s.?

Assume that we have some equality, $$ X (t) = Y(t). \quad \quad \quad \quad \quad \quad \quad (1) $$ I imagine that if I say "(1) holds a.s. for all $t>0$" it means that $$ P\{X (t) = Y(t) \text{...
1
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2answers
41 views

Exercise about conditional probabilities

Exercise: Let $S_1$ and $S_2$ be two disjoint events, and let $\mathcal{G}$ be a sub-$\sigma$-algebra. Show that the following hold with probability $1$. \begin{align*} \mathrm{(a)}& \quad 0 \...
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0answers
55 views

Big O in Stochastic Sense

I understand that if for a real-valued random variable $X$ we have $X = O_p(1)$, then it means that for any $\epsilon>0$, there exists a positive real number $M>0$ such that $p(|X|>M)<\...
4
votes
1answer
58 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
1
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2answers
56 views

Relationship between covariance matrix and its determinant

Let $X=(X_1, X_2,..., X_n)$ be random variables $$ v_{ij} = cov(X_i, X_j) = E(X_i, X_j) - E(X_i)(X_j) $$ Show that the det of v is zero iff there are $a_1, a_2,..., a_n $ and b such that $$ P(...
3
votes
1answer
54 views

$F\big( g(t) \big) - F\big( g(t + h) \big) \leq h$ implies that $g$ is right-continuous?

Suppose $F$ is a continuous, strictly increasing distribution function. Also, suppose that $g:[0,1] \longrightarrow [0,1]$ such that for any $t \in [0,1]$, $h > 0$, and $\epsilon > 0$, $$ F\big(...
1
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1answer
64 views

Complicated Conditional Probability

Three people role a die starting with person 1, then 2, then 3. First to role a 6 is eliminated. Find the probability that B is elimination first. I think this is some summation of probabilities to ...
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3answers
198 views

Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
2
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0answers
52 views

Bounding the density of random variable

This is a followup to the question in Bounding the Density of the Maximum of N Random Variables I have a random variable, X, whose cdf is bounded as below: $ \Pr \{X \le x \} \le \underset{i}{\prod}...
1
vote
1answer
47 views

Positive submartingales

Let $\{X_n\}$, $n>0$ be a positive submartingale with $X_{0} = 0.$ Let $V_n$ be random variables such that $V_n \in\mathcal F_{n−1}$ for all $n \geq 1$. $B > V_1 > V_2 > \dots > 0$ ...
1
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0answers
51 views

Prove that $E_x[T_y]<\infty$ for irreducible positive recurrent Markov chains.

This is an exercise from Durret's "Probability: Theory and Examples". Suppose $p$ is a irreducible and positive recurrent Markov chain. Then $E_x[T_y]<\infty$ for all $x,y$. I had the following ...
0
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1answer
33 views

How to show the following about expectations

If $X$ is $\mathcal{M}_1$ measurable on a probability space, $\mathcal{M}_2\subseteq\mathcal{M}_1$ and $Y$ is $\mathcal{M}_2$ measurable and they satisfy $E(XI_A)=E(YI_A)$ for every $A\in\mathcal{M}_2$...
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2answers
51 views

Conditional probability for a RV with exponential distribution

Let X be a positive random variable such that for all $x,y>0$ we have that $$\mathbb{P}[X >x+y | X>x] = \mathbb{P}[X > y]$$ I need to show that X has exponential distribution, i.e, $\...
0
votes
1answer
74 views

Are my calculations using Neymann Pearson lemma correct?

I read this post, but I need to use N-P lemma to verify hypothesis doing it really step by step, so please help me. $X_1,X_2,\ldots,X_{30}\sim N(\mu, 1)$, so $\sigma=1$ (I assume that) and $n=30$. $...
5
votes
3answers
217 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
1
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0answers
16 views

If X~N(0,1) and Y~N(0,1) then X+Y~N(0,2) [duplicate]

I am trying to show that If X~N(0,1) and Y~N(0,1) then X+Y~N(0,2), where X and Y are independent random variables. Any help will be appreciated.