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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3
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0answers
29 views

Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq ...
-2
votes
0answers
37 views

What is the distribution of $X$? [on hold]

Let $Z=\left\{a,b,c\right\}^{\mathbb {Z}} $ and equip $\mathbb Z$ with the product topology generated by the cylindersets. On this product topology let $P$ be the product measure giving each $a$, ...
2
votes
4answers
84 views

What does this definition mean: $F_Y(y) =P(Y<y)$?

I am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means. For instance the following question: Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x ...
2
votes
3answers
76 views

What is meant by $P(X = x)$?

What is meant by the statement $P(X = x) = \theta$? As in, what is its English translation? I'm assuming that $X$ is a random variable and $x$ is a member of its sample space. Is it just "the ...
1
vote
0answers
19 views

Marginal probability from probability of a sum of random variables

If we have the probability of sum of two random variables $$\mathbb{P}(X+Y \leq \theta) $$ How can we obtain the marginal probability of $X$ My Solution I perform marginal probability as follows: ...
2
votes
0answers
38 views

The polynomial is dense in $L^2$ with non-lebesgue measure

Assume the function $u\to \mathbb E[e^{iuX}]$ is analytic in a nbhd of $0$ where $X$: $\Omega\to \mathbb R$ is a random variable. Now I want to conclude that the space of polynomial, denoted by ...
3
votes
1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
2
votes
1answer
25 views

Computing Distribution of Conditional Expectation of Gaussian RV

I am trying to compute distribution of the following random variable \begin{align*} E[(X-E[X|Y])^2|Y] \end{align*} where $X \sim \mathcal{N}(0,\sigma^2_x)$ and $Z \sim \mathcal{N}(0,\sigma^2_Z)$ where ...
2
votes
0answers
31 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
-1
votes
0answers
25 views

absolute deviation for binomial distribution [duplicate]

Let $X_{1},X_{2},...,X_{n}$ be independent Bernoulli trials being a $1$ (success) with probability $\frac{1}{2}$ and $0$ otherwise. Let $$X=\sum_{i=1}^{n} X_{i}$$ be the binomial random variable with ...
1
vote
2answers
21 views

Why do We Refer to the Denominator of Bayes' Theorem the “Marginal Probability”?

Consider the following characterization of Bayes' Theorem: Bayes' Theorem: Given some observed data $x$, the posterior probability that the paramater $\Theta$ has the value $\theta$ is $p(\theta \mid ...
1
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0answers
17 views

How to solve for a Phase Function Cumulative Distribution function (CDF) calculation give a pdf …

I am attempting to solve for the CDF (more specifically the inverse CDF, but that is easy once I have the CDF) - Cumulative Distribution function given a Probability Distribution Function (pdf) and g ...
2
votes
0answers
15 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
2
votes
1answer
41 views

If $Y$ is determined by and independent of $X$ then $Y$ is deterministic

I'm working on the following exercise: Let $X,Y$ be random variables. Show that if $Y$ is simultaneously determined by $X$ and independent of $X$ then $Y$ is deterministic. Here $Y$ is said to ...
1
vote
1answer
19 views

Bayes' Rule where the probabilities are taken as conditional

I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below. ...
2
votes
0answers
10 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
6
votes
1answer
127 views

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right)|\mathcal F_t^X\right)$

Framework: Consider a continuous stochastic process $(X_t)$ together with a Brownian motion $(B_t)$. Those two stochastic processes are assumed to be independent. Denote by $(\mathcal F_t^X)$ and ...
4
votes
1answer
29 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
3
votes
0answers
21 views

Expected value of sorted subsequence

Consider the following discrete random variable: Given an array of size n containing random unique integers, what is the maximal length of a sorted subsequence from that array. What is the expected ...
2
votes
2answers
73 views

Why is it impossible to hold these probability beliefs?

Why is it impossible to hold these probability beliefs? \begin{align*} P(a) & = 0.3 & P(a \land b) & = 0 \\ P(b) & = 0.4 & P(a \vee b) & = 0.8 \end{align*} I know that you ...
2
votes
2answers
85 views

Interesting variant of binomial distribution.

Suppose i had $n$ Bernoulli trials with $X_{i}=1$ if the $i$th trial is a success and $X_{i}=-1$ if it is a failure each with probability $\frac{1}{2}$. Then the difference between the number of ...
5
votes
1answer
64 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
4
votes
4answers
136 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
65 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
77 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
77 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
44 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 ...
2
votes
0answers
65 views
+50

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
34 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
39 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 ...
1
vote
0answers
24 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
42 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
33 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
67 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
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0answers
31 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
94 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 ...
0
votes
0answers
25 views

Show that $P(\Omega)$ is a $\sigma$ algebra [closed]

I hope you all are doing great. I have the idea of how the proof goes, but i can't show a formal proof that the union axiom is satisfied Thanks for the help!!
4
votes
3answers
122 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
16 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. ...