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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4
votes
1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
1
vote
1answer
16 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
4
votes
1answer
45 views

conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
0
votes
1answer
25 views

Convolution of a function with itself n times convergence to bell curve

If we have a piecewise function defined as $f(x) = \begin{cases} 1, & \text{0 $\le$ $x$ $\le$ 1} \\ 0, & \text{otherwise} \end{cases}$ Explain how the convolution of $f$ with itself for $n$ ...
1
vote
2answers
23 views

Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused ...
0
votes
0answers
17 views

Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
2
votes
1answer
30 views

determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
3
votes
4answers
82 views

Find the value of $\mathbb{E}(X_1+X_2+\ldots+X_N)$ of i.i.d random variables $X_i$s.

Let $ X_1,X_2,X_3 ,…$ be a sequence of i.i.d. random variables with mean $1$. If $N$ is a geometric random variable with the probability mass function $\mathbb{P}(N=k)=\dfrac{1}{2^k}$; $k=1,2,3,\...
3
votes
1answer
86 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
1
vote
0answers
31 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
1
vote
1answer
50 views

Expectation of $|H - T|$

Using binomial approximation to normal distribution, find the expectation of $|H-T|$ where the $H,T$ are heads and tails of a fair coin and the number of tosses is large. Can anyone please tell me, ...
3
votes
1answer
29 views

Explanation of Aumann's “agreeing to disagree” in modern notation

I'm attempting to understand Aumann's classic 1976 paper Agreeing to Disagree, which claims, under certain assumptions, that if two Bayesian agents share knowledge of each others' posteriors then they ...
2
votes
1answer
34 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
0
votes
2answers
31 views

basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
0
votes
1answer
10 views

Notation clarification in Schilling's Brownian Motion

In Chapter 1's Problems, Problem 1(b), we are given that $X,Y\sim \beta_{1/2}:=\frac{1}{2}(\delta_0+\delta_1)$ are Bernoulli random variables. How am I to interpret this? That the probability of ...
4
votes
1answer
989 views

Inverse Mills ratio for non normal distributions.

We have the well known result of the inverse Mills ratio: $$ \mathbb{E}[\,X\,|_{\ X > k} \,] = \mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
0
votes
1answer
48 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...
1
vote
1answer
22 views

Find variance of happy people sitting at $n$ regular polygon table

Let table has shape of $n$ regular polygon and at each side is sitting one person. Each person is flipping a fair coin once (results of $n$ independent tosses are independent). Person is happy iff he ...
2
votes
1answer
48 views

Find the limit of $P(\bar{X_n}\leq 1.8)$ for i.i.d random variables $X_i$s of known distribution

Let $X_1,X_2,…$ be a sequence of independent and identically distributed random variables with $P(X_1=1)=\frac{1}{4}$ and $P(X_1=2)=\frac{3}{4}$. If $\bar{X_n}=\frac{1}{n}\sum_{i=1}^{n}X_i$, for $n=...
2
votes
1answer
64 views

Sequence of Erdos-Renyi random graphs convergent with probability 1

Definitions Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph ...
2
votes
0answers
28 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
0
votes
0answers
11 views

CGF determines the distribution

It is well known, that if the domain of the mgf of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. However consider a Levy-...
1
vote
0answers
32 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
3
votes
0answers
16 views

Information matrix for a Student's T distribution

I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation. $$ -\operatorname{E}_{t-1} \left[ \frac{\partial^...
4
votes
1answer
40 views

What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
0
votes
0answers
9 views

hypothesis testing for 3$n$ samples where each $n$ sample has different underlying distribution

I have a question related to hypothesis testing as follows: I have $3n$ samples where the the first $n$ samples are drawn iid according to distribution $P_1$, second $n$ samples are drawn iid ...
0
votes
0answers
38 views

Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...
-1
votes
1answer
58 views

Find $P(\limsup A_n)$ given $P(A_{n+1}) \ge \frac{n}{n+1} P(A_n)$ [closed]

Consider independent events $A_1, A_2, ...$ in the same probability space s,t. $$P(A_{n+1}) \ge \frac{n}{n+1}P(A_n) \ \forall n \ge 1$$ Find $P(\limsup A_n)$. There seem to be two cases: $\exists ...
0
votes
1answer
21 views

How to model guessing?

I want to model the knowledge of the student $i$, in a particular subject $S$. I give him a set of questions $Q$ from $S$ to test his knowledge. The level of his knowledge depends on the number of ...
0
votes
0answers
27 views

Does the pgf uniquely determine the distribution? [on hold]

I know that the characteristic function of a random variable uniquely determines the distribution, but I'm just curious whether the probability generating function does so too(assuming that it exists)....
0
votes
2answers
21 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
1
vote
1answer
35 views

Define the function $g (y) = E[f(X,y)]$. Show that $g$ is Borel-measurable, and that $E[f (X,Y)|Y=y] = g(y)$

The original question is the number 10.6 of this pdf: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a bounded Borel-measurable function, and let $X$, and $Y$ be independent random variables. Define ...
1
vote
0answers
18 views

Find conjugate prior of an exponential family distribution

I read on Wikipedia that all exponential family distributions have conjugate priors. I have not, however, been able to find a reference that describes how to find it. So given $$f_X(x\mid\theta) = h(x)...
0
votes
1answer
25 views

Khinchin's Law of Large numbers proof unclarity.

This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\...
3
votes
1answer
85 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
-1
votes
2answers
51 views

What is the variance of the following random variable? [closed]

There is a box with $5$ rows and $5$ columns, and it contains $25$ pieces. Accidentally, it falls, and the pieces are put back into the box. The number of pieces that will end up in the right place is ...
2
votes
1answer
35 views

Non-negative, integrable random variables which converge in probability and whose expected values have a finite limit

Suppose we have a sequence $X_1, X_2,...$ of non-negative, real random variables (not necessarily increasing) in $L^1$ which converge in probability to an integrable, non-negative random variable $X \...
0
votes
1answer
15 views

What is the chance of collision of latter group of card number among different people?

Let's assume we are a payment system issuing 16-digit cards. If we have X customers and issue Y cards, how to calculate the chance of at least single collision of last 4 digits within a single ...
0
votes
2answers
25 views

statistics- probability question [on hold]

Let E be the event that a corn crop has an infestation of ear worms, and let B be the event that a corn crop has an infestation of corn borers. Suppose that P(E) = 0.24, P(B) = 0.16, and P(E and B) =...
5
votes
0answers
50 views

Inverse image is $\sigma$-algebra

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$. How do I see that $\...
3
votes
1answer
27 views

Lebesgue-Stieltjes measure corresponding to a right continuous increasing function, $m(\{x\}) = \alpha(x) - \alpha(x-)$ for each $x$

Let $m$ be Lebesgue-Stieltjes measure corresponding to a right continuous increasing function $\alpha$. How do I see that for each $x$, we have$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
0
votes
3answers
79 views

How can I compute $\mathbb{E}[Z^4]$ where $Z\sim N(0,1)$

Let $Z\sim N(0,1)$ and $Y=a+bZ+cZ^2$. I want to compute the variance of $Y$. This is what I did: $$\operatorname{Var}(Y)=0+b^2\operatorname{Var}(Z)+c^2\operatorname{Var}(Z^2)=b^2+c^2\operatorname{Var}...
1
vote
0answers
61 views

For Which General Distributions Does This Inequality Hold?

Let $X$ be a random variable with mean $\mu$, where $0 < \mu < 1$. Let $X(n)$ be the sum of $n$ independent ,identically distributed, $X$ variables. Under what conditions on $X$ , possibly ...
3
votes
0answers
37 views

Knight (Chess) Problem on telephone keyboard

There is phone keyboard with Knight on 0 (as shown below). 123 456 789 0 Knight moves as per the rules of chess (2 straight and one turn). T is no. of moves ...
3
votes
1answer
63 views

The Uncountable and Probability

Suppose we draw a random uniformly number from $[0,1]$, if we do this countable many times, how many times will we get $1$, I suspect $0$? If we do it uncountable many times, how often will we get $1$?...
0
votes
1answer
25 views

What does a random variable 1 with subscript [0,1/2] mean?

I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure) It is hard to google this, too. What does such ...
1
vote
2answers
33 views

Is there a book or lecture notes on Percolation Theory containing exercises?

I have seen Grimmett's Percolation Theory and I have also seen a few online lecture notes. But they don't have exercises. I understand it is stupid to ask of exercises in such a recent and hot ...
3
votes
0answers
34 views

Showing that $\sigma$-algebra is uncountable [duplicate]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
0
votes
2answers
21 views

Probability of two statistically independent, uniformly distributed variables occurring within time frame of each other?

Say two events will occur independently of each other, only once each. The time of each event occurring is uniformly distributed from 0 to 10 seconds. What is the probability that the events will ...
2
votes
1answer
21 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...