Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

learn more… | top users | synonyms (1)

0
votes
1answer
16 views

The probability that $3$ random points on the circumference form a right-angled triangle?

In my probability theory course, I dealt with a similar problem which asks for the probability that $3$ random points on the circumference of a circle lie on the same semi-circle. But it makes me ...
0
votes
0answers
9 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
0
votes
0answers
17 views

Proving that Poisson distribution is well-defined

I'm trying to prove that a Poisson distribution is a well-defined probability distribution -- i.e. that the sum of probabilities over all possible values is one. Since the distribution takes on ...
0
votes
2answers
14 views

is it true that conditional expectation Y to X is a function of X?

I mean, is it true that $E(Y|X) = \phi(X)?$ if so, how should we derive the form of X?
3
votes
0answers
19 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
1
vote
1answer
17 views

Expectation over 2 random variables, help needed

Hi I am new here and I hope I can get some help. My question is about taking expectation over random variables. Lets say I have two random variables $\Xi$ and $\theta$ where $\Xi$ is for example a ...
0
votes
1answer
39 views

In frequentism, does every event have a probability?

For an infinitely repeatable trial with event space $\Omega$, and an event $A\subseteq \Omega$, the frequentist probability of $A$ is defined: $P(A):= \lim_{n\rightarrow\infty} \frac{n_a}{n}$, where ...
0
votes
1answer
11 views

Probability of non repeated value in a set of vectors (with integer values) for any number in the same vector position.

Suppose a set with $m$ vectors ($m$ finite) defined by $V_{i} = (x_{vi1},x_{vi2},\dots,x_{vin})$, with $i \in \left\{1, 2, \dots, m \right\}$ and $2 \leq n \leq p$, for a given $p \in \mathbb{Z}$ ...
1
vote
2answers
28 views

Optimal Number of White Balls

There are C containers, B black balls and infinite number of white balls. Each container should have at least one ball. Each of the containers may contain any number of black and white balls. Action ...
0
votes
0answers
14 views

Help me understand how finding related distributions work

$$G(a)=\frac{6400}{a^2}$$ So this is the question, and I know well to answer any type of question like this. Here's how I do? (i) $$F(t)=\int_{1/2}^{T} \frac {1}{2t^3} dt = ...
1
vote
1answer
26 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
0
votes
1answer
23 views

Bounding the difference of random variables by coupling.

Suppose we have two probability densities differing by atmost $\delta$. Is it possible to use coupling to have two random variables with the above two densities differing by less than $\delta$? I ...
0
votes
0answers
35 views

$\tau$ is stopping time. Check if $\tau + 1$, $\tau - 1$, $\tau^2$ also are stopping time.

Suppose that $\tau$ is stopping time. Is is true that a) $\tau + 1$ b) $ \tau - 1 $ c) $ \tau^2 $ also are stopping time? My prove: a) Yes, because forall t we have $$\{ t: \quad \tau+1 \le t \} ...
0
votes
2answers
16 views

Inner dependence of Independent random vectors

If $X = (X_1,X_2)$ and $Y = (Y_1,Y_2)$ , $X$ and $Y$ are stochastically independent can $X_1$ and $Y_1$ be dependent?
0
votes
0answers
19 views

Find $E[Z_1 | aZ_1 + bZ_2]$

Let's $Z_1,Z_2$ be a random variable such that $EZ_1^2 < \infty$ and $EZ_2^2 < \infty$. Find $E[Z_1 | aZ_1 + bZ_2]$ where $a,b \in \mathbb{R}$. We don't know what is distribution of $Z_1$ and ...
0
votes
0answers
14 views

expectation approximation error

Let $X$ be a random variable with no mass taking values in $\mathbb{R}$, and $f:\mathbb{R}\mapsto\mathbb{R}$ be a "smooth" function. I want to approximate $\mathbb{E}[f(X)]$ with $\mathbb{E}[g(X)]$ ...
0
votes
1answer
18 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
2
votes
0answers
26 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
1
vote
1answer
22 views

Simple bounding question for an expectation with truncating function

Let $\{X_m\}$ be independent random sequence. I want to show the following result Given $E[X_m^2]:=\sigma^2 < \infty$ and $$0 = \mathop {\sup }\limits_m P\left( {\left| {{X_m}} \right| > ...
1
vote
1answer
35 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
1
vote
1answer
45 views

Requesting deeper understanding of binomial coefficient

I noticed that $\binom {52} 4$ * $\binom {48} 1$ is $5$ times that of $\binom {52} 5$. So for example, if we were to draw $4$ cards from a standard deck then draw $1$ more, we cannot just say there ...
0
votes
0answers
16 views

Probability distribtuions [on hold]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
0
votes
1answer
32 views

$X$ normally distributed in $\mathbb R^n$ iff components $x_i$ normally distributed?

We've had the normal distribution today in class and I was thinking about the following: Let $X$ be normally distributed, $X\sim N(a,\Sigma)$ with a symmetric positive definite matrix $\Sigma$ and ...
0
votes
0answers
18 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
0
votes
0answers
16 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
1
vote
1answer
8 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
0
votes
0answers
16 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
0
votes
2answers
11 views

Conditional probability: $P(B'|A) = 1-P(B|A)$

Suppose that $A$ and $B$ are events with $P(A) > 0$. Show that $$P(B'|A) = 1-P(B|A),$$ where $B'$ is the complement of $B$. I get stuck after I go from $P(B'|A)$ to $P(AB')/P(A)$. I would greatly ...
1
vote
0answers
26 views

Law of iterated logarithm proof

I am trying to master this proof of iterated logarithm. However, I get stuck at the last part. Here is a link In the last two line at fourth page. We calculate the probability that: $$ (*) ...
2
votes
0answers
32 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
0
votes
0answers
71 views

An elementary annoyance

I'm going through some notes and I'm having problem understanding an inequality: The objects involved are: $X$ is a real-valued random variable with mean zero. We consider $n$ identical copies ...
1
vote
1answer
10 views

probability theory: best predictor allowing for nonlinear predictors

In chapter 7 the optimal linear predictor of Y based on $X = x_i$ was found. The criterion of optimality was the minimum mean square error, where the mean square was defined as $E_{X, Y}[(Y - (aX + ...
0
votes
0answers
6 views

average distance between vectors of n dimenstions [on hold]

In a recent experiment I did, I observed that the minimum euclidean distance for vectors of about 10k dimensions (each single feature is has standard normal distribution) is about 20, even if I sample ...
0
votes
2answers
30 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
1
vote
1answer
27 views

How to calculate $\mathbb{P}[Y\in F|X]_{\omega}$

Here I have an exercise of book: Probability and Measure of PATRICK BILLINGSLEY of conditional probability in the page 442, exercice 33.4 (b): Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
1
vote
2answers
37 views

Why is the expected value of $|X|^p$ equal to $p\int_{0}^{\infty}y^{p-1}\mathbb{P}(|X|>y) dy$?

I'm trying to understand a passage from the book: A Basic Course in Probability Theory, Rabi Bhattacharya Edward C. Waymire, in the page 21. The calculation is the following: If $X$ is a random ...
0
votes
0answers
25 views

Sigma algebra generated by a random vector

I understand this question is very basic, but I found this confusing while I am learning measure theory myself.. Suppose we toss a coin twice (once afeter once), and denote by each $X$ and $Y$ the ...
0
votes
0answers
17 views

Showing a fact about $\sigma$-algebras and Borel sets

Let $(\Omega,\mathcal{A})$ be a measurable space, $(A_n)_{n\in\mathbb{N}}\subset\mathcal{A}$ and $f_n:\Omega\to [-\infty,\infty]$ be a $\mathcal{A}-\overline{\mathcal{B}}$ measurable function, where ...
1
vote
1answer
27 views

Proving (a part of) Hoeffding's lemma

Hoeffding lemma goes like this: *Let $X$ be a scalar variable taking values in an interval $[a,b]$. Then for any $t>0$ $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} ...
1
vote
0answers
18 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
0
votes
0answers
18 views

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$ $D$ is a set of discontinuous points X and $f$ is bounded, measurable. We can ...
1
vote
0answers
10 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
0
votes
2answers
31 views

theory of probability question [on hold]

There is a lottery. From 10,000 people only 100 win, so the probability to win is 1%. Question: what is the probabily to win if you join/buy ticket to the same lottery 100 times? I am sure you can ...
3
votes
2answers
32 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
1
vote
1answer
21 views

Is an infinite product probability space compatible with an almost surely statement?

My question pertains to the following Theorem which can be found here. Theorem (Existence of product measures): Let $A$ be an arbitrary set and for each $\alpha \in A$ let ...
2
votes
2answers
31 views

Relationship “finite mean” <-> "absolutely integrable

What is the relationship between the property of a random variable (i.e. a measurable function defined on some probability space) being absolutely integrable, i.e. $$\mathbb{E}|X|<\infty$$and ...
0
votes
0answers
43 views

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? [duplicate]

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? This is not a homework problem but rather a question I had. If it is not true, what are the ...
1
vote
0answers
32 views

Application of Law of Iterated Expectations

Could you please explain something from a text I am reading? We're given that $E(\epsilon_i)=0$ and $E(\epsilon_i x_i)=0$ for $i\geq 1$. Write $g_i=\epsilon_i x_i$ and we further know that $$ ...
2
votes
1answer
45 views

Simple Question about Almost Sure Convergence

If I can show for some event $A_n$, $$ \sum_n P(A_n) < \infty. $$ Then by First Borel-Cantelli Lemma, I get $P(A_n \; i.o.) = 0$; But I still confused about how this infinitely often connects to ...
0
votes
0answers
12 views

Conditional expectation, conditioning on a random variable [duplicate]

I am asked to show that if $X,Y$ are integrable, and $E[X| Y]=Y$ and $E[Y|X]=X$ almost surely, then $X=Y$ almost surely. Moreover, is the first equality sufficient for $X=Y$ almost surely? My attempt ...