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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2
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1answer
29 views

Joint Distribution Implies Independence…?

Consider the measurable space $(\mathbb{R}, \mathcal{B})$ and a probability space $(\Omega, \mathcal{F}, P)$. Define a finite sequence of random variables $X_1,\ldots,X_n: \Omega \to \mathbb{R}$. ...
0
votes
1answer
10 views

Equivalence of definitions of Gaussian Measure

Wikipedia's article on Gaussian measures notes this as the definition of Gaussian measures: $\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 ...
1
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0answers
10 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
1
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1answer
36 views

Trouble finding the expected value of a random variable

Suppose that we have a procedure A that we run once and it returns as a result either success or ...
0
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0answers
14 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
2
votes
1answer
22 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
0
votes
0answers
19 views

A conjecture on the connection between the difference of two independent Poisson random variables and their parameters.

Let $X$ and $Y$ be two independent poisson random variables with parameters $\mu$ and $\lambda$, respectively. Assuming that $\mu\geq\lambda$ , is it true that $P\left(X=Y-k\right)$ is decreasing in ...
-3
votes
0answers
32 views

Is a limit of measure of a sequence of sets equal to measure of limit of the sequence of sets?

I'm sitting at the same question desk as this: Limit of the measure of the converging sequence of sets. Actually, I can't prove it neither. PA6OTA gave a hint to show there is subsequence $A_{n_k}$ ...
-2
votes
1answer
21 views

Find probability mass function from text

$5$ persons (each independent of the other) when in a good mood it opens the tap with probability $\frac{1}{2}$ or in a bad mood with probability $\frac{1}{2}$. When that person is in a good mood it ...
1
vote
1answer
21 views

Normally Distributed = Absolute Continuity?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable $X: \Omega \to \mathbb{R}$ is said to have the standard normal distribution if it has the density $f:\mathbb{R} \to ...
0
votes
2answers
36 views

How to find E(Y) given that the random variable X is exponentially distributed with lambda equal to 0.5?

Random variable $X$ is exponentially distributed with the parameter $\lambda$ equal to $0.5$. Define also $Y = 1 - 2X$ Find $E(Y)$ , Var(Y) and the moment generating function of Y. I have $f_x(X)= ...
1
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2answers
36 views

How is the definition of joint probability not circular?

According to the chain rule: $$ P(A,B) = P(A \cap B) = P(A \mid B)P(B) $$ Yet, according to wikipedia, the Kolmogorov definition of conditional probability is (axiomatically) as follows: $$ P(A ...
2
votes
1answer
42 views

Convergence of a sequence

Let $X$ be a random variable with a distribution function such that $n^t P(|X|>n) \to 0$ as $n \to \infty$, for some $t>0$. Then, I know that for any $\epsilon>0$, there exists $n_0\in ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
5
votes
1answer
60 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
2
votes
1answer
84 views

What distribution has $X^n$ if $X$ is normal distributed?

Let $X$ be a random variable with mean $0$ and variance $\sigma ^2$, i.e. $X \sim \mathcal{N}(0, \sigma ^2)$.What is the distribution of $Y= X^n$, $n \in \mathbb {N}.$ ? I know what distributribution ...
0
votes
0answers
32 views

Does it follow that $P(X_n(x_1)=0,…,X_n(x_k)=0)\to 1$?

Let $X_n(x)$, $x\in\mathbb{Z}$, $n=1,2,\ldots$ be random variables. I know that for all $x\in\mathbb{Z}$, $$ P(X_n(x)=0)\to 1\text{ as }n\to\infty. $$ Does it follow for $x_1,...,x_k\in\mathbb{Z}$ ...
2
votes
2answers
59 views

A Taylor series expansion of $e^{ix}$

In Probability Theory by Athreya and Lahiri, they give a very elegant proof of Central Limit Theorem (The Lindeberg one) wherein they use a lemma: For $x \in \mathbb{R}$ and $r \geq 1$, ...
0
votes
0answers
21 views

An inequality of probability [on hold]

Suppose $X$ is a random variable. $E(X^2)=1$, $E|X|\ge a>0$. Let $\lambda\in[0,1]$, prove that $$P(|x|\ge\lambda a)\ge(1-\lambda)^2a^2.$$ I have tried to use Chebyshev's inequality, but I didn't ...
1
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0answers
28 views

Why is this definition of the mesaure $\mu$ not properly?

Consider the measurable space $$ (\left\{a,b,c\right\}^{\mathbb{Z}},\mathfrak{A}), $$ where $\mathfrak{A}$ is the product topology on $\left\{a,b,c\right\}^{\mathbb{Z}}$ which is generated by the ...
1
vote
1answer
37 views

What is Skorohod's Represntation Theorem Saying?

From Wikipedia: Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. ...
5
votes
1answer
31 views

$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
9
votes
1answer
88 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
-2
votes
0answers
31 views

Impact of perturbation on the eigen-values of 3 diagonal matrix [on hold]

Lets consider a 3-diagonal matrix as following: $$ A(i,i) = 2 $$ $$ A(i,i+1) = -1 $$ $$ A(i,i-1) = -1 $$ The eigen-values of this system is known easily. How eigen-values would change if we add ...
2
votes
0answers
22 views

Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
1
vote
1answer
37 views

Definition of a discrete random variable

Here is the defintion of discrete random variable from "An introduction to probability and statistics" by Rohatgi. Let $(\Omega,S,P)$ be a probability space. An random variable $X$ defined on this ...
2
votes
0answers
19 views

Distribution of $f(x,|h|)$, being $|h|$ rayleigh distributed

INTRODUCTION Let's supose we receive the following signal: \begin{equation} y[n] = hx[n]+W[n] \end{equation} where: $x[n] = Ae^{j2 \pi f_c t}$ is the transmitted signal $f_c$ is the carrier ...
-3
votes
0answers
35 views

How to compute P(Y>X)? [on hold]

Compute $P(Y>X)$ when $F_{X,Y}(x,y)= \begin{cases} 3\over4& \text{if }(0< x < 2) \wedge (0 < y < 2x-x^2)\\ 0 &\text{elsewhere}\end{cases}$ $F_X(x)=\begin{cases}\frac32 ...
1
vote
0answers
17 views

Random variable with respect to the event space [on hold]

Having the probability space $(S, \mathcal F, \Pr(\cdot))$ and a very large $\mathcal F$, like the power set, how do we define a function $X(\cdot): S \rightarrow \mathbb{R}$ which is not a random ...
-2
votes
0answers
57 views

How to solve for X^2-2Yx+Y=0? [on hold]

How can I solve for $x^2-2Yx+Y=0$? Note: Y is an exponentially distributed random variable with parameter lambda>0. The solution is the following: no real solution for $4Y^2-4Y<0$, so when ...
2
votes
1answer
29 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
3
votes
0answers
27 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
83 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
36 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
57 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 ...
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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
40 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
18 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
122 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
28 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 ...