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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
19 views

A question about distributions/densities

Given two random variables $X,Y$ how to show that $P(X\leq Y+x)=\int F_X(y+x)f_Y(y)dy$? I know that $f_Y(y) = \int f_{XY}(x,y)dx$, but have no idea how to go with the previous equation.
4
votes
1answer
142 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
2
votes
2answers
114 views

Why is this standard deviation $20$?

If I have two random variables $X_1$ and $X_2$ with $X_1\sim N(520,10)$ and $X_2\sim N(500,10)$, and $X_1$, $X_2$ are both speeds of airplanes where the first one is 10 km ahead of the second one. I'm ...
3
votes
2answers
86 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
0
votes
2answers
697 views

How to convert a histogram to a PDF

I know this may be an easy question, but due to lack of math knowledge I do not know the answer. Would you please explain to me with a simple example that how can I find PDF from a histogram. Thank ...
3
votes
1answer
42 views

Independent sets in subfield

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $\mathcal F$. Suppose that $A_1,\ldots,A_n$ are independent sets belonging to $\mathcal F$. Let ...
3
votes
2answers
34 views

Expectation of the squared error with regards to a sub sigma field

I am totally stuck. Given a probability space $(\Omega, \mathcal F, \mathbf P)$ and a random variable $X$. Let $\mathcal A$ be a sub-$\sigma$-field of $\mathcal F$. Let $Y$ run over all $\mathcal ...
4
votes
3answers
2k views

Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable

Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X>k). $$ How to prove this? I tried a bit but unable to post due to formatting ...
1
vote
1answer
185 views

Mean and variance of geometric function using binomial distribution

Can anyone help solving this question please? I tried but not sure of the steps to reach the conclusion.
1
vote
1answer
120 views

polynomial approximation on compacts

Let's say $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is of class $C^k$ with $k \geq 0$. How do I know that I can find a sequence of polynomials such that all its derivatives up to order $k$ converge ...
2
votes
2answers
73 views

How do I approximate this binomial problem correctly?

If I have an experiment that has $1000$ trials, and $10$% of the time there is an error, what is the approximate probability that I will have $125$ failures? I figured out that $\mu =100$ and $\sigma ...
0
votes
1answer
134 views

Application of Markov's inequality

let $h\colon \mathbb{R} \to [0,\alpha]$ be a nonnegative bounded function. Show that for $0\leq a<\alpha$ that the following holds: \begin{equation} Pr(h(X)\leq a) \geq \frac{E[h(X)]-a}{\alpha-a} ...
2
votes
0answers
110 views

Textbook Recommendation; Proability Theory with Measure Theory

I'm currently taking a course in Probability Theory and was hoping someone could point me in the direction of a useful supplementary textbook. Our course currently uses A Modern Approach to ...
5
votes
1answer
772 views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
0
votes
1answer
427 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
2
votes
1answer
238 views

Convergence of sum of product of i.i.d. random variables

Consider a sequence of i.i.d. random variables, $(X_i)_{i=1}^n$, with zero mean and unit variance. I want to calculate the limit (a.s.) of $$ \frac{1}{n}\sum_{i\neq j}X_iX_j $$ as $n\to\infty$. ...
1
vote
1answer
607 views

Borel sigma field

Is the $ \sigma $ - field generated by $[a,b]$, $a,b \in \mathbb Q $ and $ \sigma $ - field generated by $(a,b)$, $a,b \in\mathbb Q^c $ identical? Are they also the same as Borel $\sigma $ - field. ...
1
vote
0answers
53 views

Given sample space and distribution, how to construct a random variable with the same distribution?

Let $(S,d)$ be a metric space, $\sigma(S)$ is generated by metric topology of $S$.$([0,1], \mathcal{B}([0,1]), \bf{P})$ is the sample space. $\bf P'$ is a probability measure on $(S,\sigma(S))$. Is it ...
1
vote
1answer
746 views

Is maximizing entropy equivalent to minimizing the defined variance?

Assume there is multi-set of some integers : $D = \{a_1,a_2,\cdots,a_{N-1}\}$ such that $\sum_i a_i = A$ we can build a discrete probability distribution by dividing elements of set by $A$, i.e. $p_i ...
5
votes
2answers
2k views

What is the PDF of random variable Z=XY?

Given two independent random variables X and Y, how can I find the PDF of random variable $Z=XY$? *If their joint distribution is required, assume that we also have it.
4
votes
1answer
1k views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
4
votes
2answers
191 views

Show $(\Omega, \mathscr{F}, P)$ is a probability space.

Show $(\Omega, \mathscr{F}, P)$ where $\mathscr{F}=\{\text{all subsets of }\mathbb{R}\text{ such that either }A^c\text{ or }A\text{ is a countable set}\}$, and $P(A)=0$ if $A$ is countable, $P(A)=1$ ...
3
votes
2answers
257 views

Expectation of Log of a Cauchy-distributed Random Variable

I found this in an article, but I cannot follow the step to get $\mathbb E[\log |a_{N,k}|]$. I'm quoting the paper: Let $a_{N,k}$ be Cauchy-distributed random variables with parameter $N(k+1)$. The ...
-1
votes
1answer
42 views

Combining output of two machines

Two machine are making some products, with machine $1$ producing twice as many items as the machine 2. However, about $4\%$ of the items from the machine $1$ and about $2\%$ items from the machine ...
21
votes
1answer
716 views

Slowest frog on a ladder amongst many, how fast does it climb and how much is it lagging below the others?

In English: Frogs are climbing up a ladder. Each frog jumps to the next level of the ladder at unit rate and independently of the other frogs and of the level it is at. All the frogs start at level ...
1
vote
0answers
113 views

Relation between Strong and Weak Law of Large Numbers

I am trying to prove the following theorem somewhat indicating the relationship between Strong and Weak LLNs: Let $\{S_n\}$ be the partial sums of a series of independent r.v.'s $\{X_n\}$. Then ...
0
votes
3answers
91 views

logic problems interesting

There is a treasure chest with 0-1000 coins in it, uniformly distributed. You place a bid on the chest and if 1.5 times your bid is greater than whats in the chest, then you get chest. What is your ...
1
vote
1answer
245 views

Interesting probability expectation question

Six individuals, including $A$ and $B$, take seats around a circular table iat random. Suppose the seats are numbered $1,...,6$. Let $X=A$'s seat number and $Y=B$'s seat number. If $A$ sends a written ...
7
votes
2answers
61 views

Interesting but difficult board game question?

I have a 3 by 3 board game. A black marble is randomly place in one of the nine squares. Distance between squares is measured as one if either diagonal or horizontal/vertically next each other, and ...
1
vote
1answer
306 views

Show that the max of two submartingale is also a submartingale

Show that the maximum of two submartingales (relative to the same filtration) is a submartingale.
2
votes
1answer
143 views

Expectation conditioned on a sub sigma field

Let $X$ and $Y$ be two integrable random variables defined on the same probability space $(\Omega,\mathcal F,\mathbf P)$ Let $\mathcal A$ be a sub-sigma-field such that X is $\mathcal A$-measureable. ...
1
vote
2answers
158 views

Delta function as a probability density distribution?

What does it mean to define a probability density function as Dirac-delta function. I mean Dirac delta is not a function itself. So how is it considered to be an example of a density function? I need ...
1
vote
1answer
30 views

How do I find this probability?

I have the joint pdf: $f(x,y)=xe^{-x(1+y)}; x,y\ge 0$ which represents the useful lifetimes of $X$ and $Y$ of a minicomputer and I am told to find the probability that the lifetime $X$ of the first ...
0
votes
1answer
61 views

Constant Density Function Property

I was wondering if, for any pdf of the type: $f_{x,y}(x,y) = c$, we can just calculate the area of integration and interpret it as the probability of the random vector. I know this would be true if ...
9
votes
0answers
251 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
1
vote
1answer
107 views

How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
2
votes
1answer
331 views

a finitely additive measure that is continuous at $\phi$ is $\sigma$-additive

Let $ (\Omega,M)$ a $\sigma$-algebra of events, and let $P$ be a finitely additive measure. We say that $P$ is continuous at an event $ A\in M$ if $A_n,B_n\in M$ are sequences of events such that ...
3
votes
1answer
55 views

Is the strong convergence of Borel probability measure metrizable?

In a metric space $(X,e)$, a sequence of Borel probability measure converges strongly, $\mu_i \xrightarrow{s} \mu$, iff for each Borel subset $S \in X$, we have $\lim_{i \to \infty}\mu_i(S) = \mu(S)$. ...
0
votes
1answer
536 views

if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$

Let $P$ be a probability measure on $(\Omega, \mathscr{F})$, If $a,b,\in\mathscr{F}$ and $A\subset B$, then $P(B)-P(A)=P(B\setminus A)\ge 0$. So far I have: $A\subset B \implies B\setminus A = B\cap ...
2
votes
0answers
89 views

Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
2
votes
1answer
52 views

If $\mu_i \xrightarrow{w} \mu \Rightarrow \int_X f d\mu_i \to \int_X f d\mu$ for all convergent sequences, does it imply $f$ is continuous?

Let $(X,d)$ be a metric space and $\mu, \mu_i $ be Borel probability measures on $X$ for $i \in \Bbb N$. If for all weakly convergent sequences $\mu_i \xrightarrow{w} \mu$ in $\Delta(X)$ we have ...
2
votes
1answer
101 views

Probability Problem with numbers

There are $n$ tickets on the table. Each ticket has a number written on it. The $i$-th ticket can be numbered $A_i$ with probability $P_i$ percent and with probability $100-P_i$ percent it can be ...
0
votes
1answer
190 views

Having trouble interpreting what the question asks about geometric distributions

I have a problem that says "Automobiles arrive at a vehicle equipment inspetion station according to a Poisson process with a rate $\alpha=10$ per hour. Suppose that with probability $.5$ an arriving ...
1
vote
1answer
232 views

Am I missing something in finding the mean and std. deviation for this lognormal distribution?

I am told that I have a random variable $X$ and the mean of this random variable is $10,281$ and the coefficient of variation is $.4$. Since we know that coefficient of variation is ...
2
votes
1answer
44 views

property of algebra

$C$ is a non-empty collection of subsets of a space $\Omega$. Let $F(C)$ be the smallest algebra containing $C$. Show that for each $B\in F(C)$, there exists a finite subcollection $C'$ of $C$ such ...
2
votes
1answer
393 views

Measurability of supremum over measurable set

Consider a finite-valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$; and a closed-valued, measurable, set-valued mapping $S: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ . Measurability is ...
-1
votes
1answer
116 views

Property of conditional expectation

It is well known that if $X, Y$ are independent random variables with $XY$ integrable, then $X$ and $Y$ are integrable and: $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$ Suppose that $\mathcal{G}$ ...
0
votes
1answer
81 views

Given density $f_U(u)$ and $U=X_1/X_2$, can we find $f_{X_1}(x_1)$ and $f_{X_2}(x_2)$?

Problem Statement: Let $X_1$ and $X_2$ be independent random variables. Given the density function, $ f_U(u)$, and the relationship: $$U=X_1/X_2$$ Can we find the density functions for $X_1$ and ...
2
votes
2answers
546 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
1
vote
1answer
326 views

Quantile function continuity

Given an increasing, right continuous function $$F:\mathbb{R} \rightarrow [0,1]$$ such that $\lim_{x\rightarrow \infty} F(x)=1 $ and $\lim_{x\rightarrow -\infty} F(x)=0$ then we can define ...