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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4
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4answers
138 views

Conditional Probability Cupcakes

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with conditional probability, which yields the shortest, simplest proofs, but ...
2
votes
1answer
66 views

unbalancing lights

I'm reading the following notes on unbalancing lights, http://www.cs.berkeley.edu/~sinclair/cs271/n5.pdf. The question i have is regarding the first page. Where it says Consider a square $n ...
5
votes
2answers
79 views

Is Keno a fair game?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other ...
2
votes
0answers
80 views

Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
0
votes
2answers
47 views

Almost sure convergence of $\max(X_1, X_2,\ldots,X_n)$.

$X_1, X_2,\ldots, X_n$ are independent with uniform distribution on $[0,a]$. Prove that $\max(X_1, X_2,\ldots,X_n) \rightarrow a$ almost surely. Ar first I look for the probability distribution i.e. ...
1
vote
1answer
44 views

Is it possible to determine if a process is random

Imagine the following experiment: someone is sitting behind the screen and calls out a sequence of numbers: "1! 3! 5! 3! 4! ...". Let's say he/she and I agreed beforehand that all numbers are ...
4
votes
1answer
98 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
0
votes
0answers
35 views

determining if it is a random variable

I know that $\int_0^{\infty}e^{-\alpha t}c(X_t)dt$ is a random variable when $c(.)$ is a measurable function and $X_t$ is a stochastic process. How can this be proved rigorously?
0
votes
1answer
42 views

Question about Independent Random Variables

Suppose that $X,Y,Z$ are independent random variables. Then is it true that $X+Y$ and $2Z$ are independent random variables? I think it is true. Here is a proof sketch of my attempt: First show that ...
2
votes
1answer
30 views

measure-preserving transformations are spectrally isomorphic

If $(X_{1}, \mathcal{B}_{1}, m_{1})$ and $(X_{2}, \mathcal{B}_{2}, m_{2})$ are probability spaces together with measure-preserving transformations $T_{1}:X_{1}\to X_{1}$,$T_{2}:X_{2}\to X_{2}$. How ...
1
vote
0answers
35 views

Limit of integrals of simple functions over a finite measure

We are given a sequence of simple functions $f_n:\mathbb{R}^2\rightarrow \mathbb{R}$ which converge pointwise to a continuous limiting function $f$. We also have a bunch of positive, finite measures ...
0
votes
1answer
36 views

Upper bounds on $E[{\rm var}^2(X|Y)]$

I am looking for an upper bound on the quantity \begin{align*} E[{ \rm var}^2(X|Y)] \end{align*} where ${\rm var}(X|Y)=E[(X-E[X|Y])^2|Y]$. Getting a lower bound is rather easy using Jensen's ...
1
vote
0answers
26 views

Define r.v. by partition of $\Omega$ and other integrable r.v., and show its integrable

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\{ A_i \}_{i\in I}$ be an at most countable partition of $\Omega$ (i.e. $I = \{1,\ldots, n\}$ or $I = \mathbb N$). Also let $X : \Omega ...
0
votes
2answers
46 views

Am I using these Probability Theory Terms Correctly?

Let $\Theta$ be a random variable with sample space $\{\theta_1, \ldots, \theta_n\}$. Questions: Is it terminologically correct to refer to $\Theta$ as a "parameter"? What about the members ...
0
votes
2answers
43 views

Computing a conditional expectation for uniform RVs

Suppose $X_1, ..., X_n \sim U[0, 1]$ are iid uniform RVs. How would I go about computing $E[X_n | X_{(n)}]$ where $X_{(n)}$ is the nth order statistic, i.e. $\max\{X_1, ..., X_n\}$ ? I'm stuck ...
0
votes
1answer
21 views

Variance of an integral of Brownian Motion

Let $W(u)$ $(u \geq 0)$ be a Brownian motion on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$. Let $I(T) = \int_0^T W(u) du$. One can use integration by parts to show that $I(T) = ...
0
votes
1answer
18 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
0
votes
1answer
34 views

an application of bounded convergence theorem

I have read the following statement: $P\{Y_{t}\neq Y_{t+u} \mid Y_0=i\}=\sum_{j\in E}P_t (i,j)[1-P_u(j,j)]$, where $E$ is a countable set. As $u\downarrow 0$ we have $1-P_u(j,j)\to 0$. This implies ...
0
votes
1answer
24 views

If $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$?

Suppose I have a sequence of random variables $X_t$ adapted to a filtration $\mathcal{F}$ when is it true that if $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$ ? ...
2
votes
1answer
68 views

Infinite Sample Space

I came across this line in a textbook I'm reading, When $\Omega$ is infinite, its power set is too large a collection for probabilities to be assigned reasonably to all its members. I'm not ...
1
vote
0answers
32 views

Conditional expectations related to Black-Scholes formula

While computing the price at time $t$ of a European call option with strike $K>0$ and maturity $T>0$ for $t$ in $[0,T]$, I encountered the following conditional expectation which I cannot ...
0
votes
2answers
95 views

Bayes' Theorem, Rigorously

For a given probability space $(\Omega, \mathcal{F}, P)$ Bayes' rule is given by $$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \quad \text{for all } A,B \in \mathcal{F}. $$ However, in many examples (in fact, ...
0
votes
2answers
41 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
4
votes
3answers
130 views

Facebook Question (Data Science)

Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview) You're about to get on a plane to Seattle. You want to know if you should bring an umbrella. You call 3 ...
1
vote
1answer
33 views

Probability inequality problem about discrete random variable

Here is the problem. Let X be a discrete random with $\ E(X) = 0$ and $\ \text{Var}(X) = \sigma^2 < \infty $. Show that $$ P(X \geq a) \leq \frac{s^2}{s^2 + a^2} $$ for all $\ a > ...
1
vote
0answers
27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
0
votes
0answers
17 views

correlated random vectors converge to an independent random vector

Suppose $(X_k,Y_k)_{k\geq 1}$ is a sequence of random vectors. $X_k,Y_k$ have support $[0,1]$ and a joint density function $ f_k(x,y)$ being positive for all $x,y\in [0,1]$ and continuous everywhere. ...
0
votes
1answer
42 views

Expected number of dice throws to fill out a table.

Say I have a table of numbers ${1,2,3,4,5,6}$. Every time I throw a fair die, if the position in the table is unchecked, it becomes checked and if it is already checked it becomes unchecked. For how ...
0
votes
0answers
21 views

Expected value of signals

I am trying to learn DPS. A couple of explanations are based on statistics. I would like to understand what is and how coherence works, but I am stuck on its definition. I found the following ...
0
votes
2answers
59 views

Random equation-does it make sense?

What is the probability that the equation $$x^2+2bx+c=0$$ has real roots? Answer is exactly $1$. (or $100$%) For example: if $b=1$ and $c=2$ roots are complex. Does it make sense? If $P(A)=0$, then ...
1
vote
1answer
26 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n ...
1
vote
1answer
28 views

Expected time to fill a table [duplicate]

Say I have a table of numbers 1-6. I throw a 6 sided die a number of times. Each time I get a number I have not already had, I mark it in the table. What is the expected number of times to throw the ...
2
votes
0answers
34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
2
votes
2answers
36 views

Var$(X) = \mathbb{E}((X - \mathbb{E}(X))^2) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$

I have a question about something my teacher told us: let $\mathbb{E}$(X) donate the expected value of a certain random variable $X$. Then Var$(X) = \mathbb{E}[(X - \mathbb{E}(X))^2] = ...
4
votes
4answers
45 views

Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$.

I'm having trouble with solving this problem: Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$. I know ...
0
votes
2answers
31 views

Take $k$ shoes ($k \leqslant n$) from a wardrobe. What is the expected value of the number of pairs ($X$) you take?

I'm having trouble with this question: Let there be $2n$ shoes ($n$ pairs) in a wardrobe, arbitrary ordened. Take $k$ shoes ($k \leqslant n$) from that wardrobe. What is the expected value of the ...
0
votes
0answers
27 views

$\lim_{n\rightarrow\infty}\operatorname{E}(X_n^2)=0$ implies $\lim_{n\rightarrow\infty}\operatorname{E}(X_n)=0$

Why $\lim_{n\rightarrow\infty}\operatorname{E}(X_n^2)=0$ implies $\lim_{n\rightarrow\infty}\operatorname{E}(X_n)=0$? Is it really that simple?
0
votes
1answer
52 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
1
vote
1answer
20 views

Determining the population size of a branching process

Suppose that I have the following branching process. Each parent can have up to two children, the number of which is determined by two independent fair coin flips. I know that this branching process ...
-3
votes
0answers
14 views

How to Randomly Generate SAT Scores in R with max and min? [migrated]

I'm trying to figure out how to randomly generate SAT scores in R by subsection. It follows the general form rnorm(n,mean,SD), but I also need to take into account that the minimum value has to be 200 ...
0
votes
1answer
18 views

In throwing 6n dice, what is the probability of getting each face n times? Use Stirling's formula to estimate this probability for a large n.

This question is taken from Probability Theory: A Concise Course by Y. A. Rozanov. My attempt at a solution is as a following: I think of $6n$ dice rolls as $n$ groups of 6 rolls. The probability ...
3
votes
1answer
80 views

Uniform distribution on a sphere

Consider the unit ball $S_n$ (centered at the origin) in $\mathbb{R}^n$ for $n \ge 1$ and a stochastic process $(X_t)_{t\ge 0}$ taking values in $\mathbb{R}^n$. Let $T = \inf\{t > 0 \colon X_t ...
0
votes
0answers
39 views

Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
0
votes
1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
0
votes
3answers
62 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity ...
1
vote
2answers
43 views

time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
0
votes
0answers
14 views

Continuous map of cadlag functions (one sided limits exist and right continuous) is cadlag

Recall that a real function $f$ is cadlag if the one sided limits $f (t^-), f (t^+)$ exist and $f (t^+) = f (t)$, i.e. $f$ is right continuous. Then is the following true? If $f$ is continuous and ...
5
votes
1answer
110 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
votes
0answers
22 views

The length of the set which will be covered by Brownian motion in a time $t$

I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable ...
0
votes
1answer
33 views

Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...