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 ...

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Should I avoid distribution functions in probability?

I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And ...
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1answer
24 views

What is application of gamma distribution on pure math or probability theory?

What is application of gamma distribution on pure math or probability theory? i saw it on several probability textbook as a definition, but it seems to me mathematician couldn't derived it if it is ...
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1answer
21 views

Composition of random variable with its distribution is uniform

I'm trying to solve the following problem (Exercise 11.13) from Probability Essentials by Jacod and Protter: Let $X$ be a random variable (on $\mathbb{R}$) with distribution $F$ that is continuous. ...
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34 views

Does a discrete random variable on a finite sample space always have an expected value?

Let $\Omega=\{A,B,C,D\}$,$\mathcal{F}=\{\emptyset,\{A,B\},\{C,D\},\Omega\}$, $P(\{A,B\})=1/2$, and $P(\{C,D\}) = 1/2$. Now $(\Omega,\mathcal{F},P)$ is a probability space. Let ...
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26 views

Convergence of random variables depending on the measure

Suppose the probability spaces $\left([0,1], \mathcal{B}([0,1],\mu_i \right)$ for $i=1,2,3$ , where $$ \mu_1 = \lambda , \ \ \ \ \ \ \ \ \ \ \mu_2 = \delta_1,\ \ \ \ \ \ \ \ \ \ \mu_3 = ...
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28 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
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1answer
40 views

Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function

I'm reading through the following theorem: Let $X=\{X_t,t\in T\}$ be a stochastic process. Then $\sigma (X)=\sigma ( \cup_{t\in T} \sigma (X_t))$ From my basic knowledge of measure theory, I ...
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1answer
22 views

If $\{X_n\}$ is a martingale, then $E[X_n-X_{n-1}]=0$

Apparently this should be quite simple, but I have been trying for a while and can't seem to get this. Let $\{X_n\}$ be a martingale, then we have: $$E[X_n-X_{n-1}]=0$$ According to some notes I found ...
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1answer
17 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
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2answers
26 views

Prove that mean independent random variables are uncorrelated

I need to prove $E(X|Y) = E(X)$ implies $E(XY) = E(X)E(Y)$ Is my proof correct? $ E(X)E(Y) = E(X) \int yf_Y(y)dy $ $= \int E(X) yf_Y(y)dy$ .... ($E(X)$ is a constant) $ = \int(\int ...
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1answer
16 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
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1answer
26 views

When will $\lim_{m \to \infty} Y_m = c \Rightarrow \lim_{m \to \infty} E[Y_m] = c$?

I am reading my probability theory textbook. In one example, the author use the strong law of large numbers to show that a random variable $Y_m$ converges to some numbers $c$. Thus, he concludes that ...
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1answer
68 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
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1answer
98 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
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1answer
19 views

Matching moments implies matching densities?

If $X$ and $Y$ are random variables with matching moments (ie: $\mu_X^i = \mu_Y^i (\forall i \in \mathbb{Z}^+)$ then are the density functions of $X$ and $Y$ identical (almost everywhere)? Idea: I'm ...
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1answer
17 views

stochastic matrix and inner product

Can a stochastic matrix be written as $V^{-1} D V $? V is an invertible matrix and D is diagonal. I think so but I can't think of a good proof. Also, the left eigenvectors and right eigenvectors are ...
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1answer
33 views

Understanding the measurability of conditional expectations

My question is about the conditional expectation of random variables with respect to a $\sigma$-algebra. I am having trouble getting an intuition behind the definitions among other things. I know ...
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2answers
26 views

Show convergence in distribution by the continuity theorem

So the problem I'm about to solve is to show that: $X \in \Gamma(a,b)$. Show that \begin{equation} \frac{X-E[X]}{\sqrt{Var(X)}} \xrightarrow{d} N(0,1) \end{equation} as $a \rightarrow \infty$, by ...
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1answer
26 views

Calculation of conditional probability

A problem as following: (from Prob, statistics, and random processes for electric engineering, p.264) If I want to find $P(Y\leq y \mid X=+1)$, it can be calculated as following: $P[N+1\leq ...
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1answer
22 views

A question about random walk similar Markov Chain

This is an exercise from Probability and Measure by Billingsley: Let $(X_n)$ be a Markov chain with $\Bbb{Z}$ as its state space and with transition probabilities $(p_{ij})$ such that $$ ...
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13 views

Understanding $O_p$ [migrated]

One thing I feel like I have never mastered is the concept of $O_p$ convergence and how to use it. I understand the basic idea and what bounded in probability means, but I always have a hard time ...
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4answers
706 views

Gambling problem

Question Robert will win $\$1$ with probability $\frac{1}{4}$, win $\$2$ with probability $\frac{1}{4}$, and lose $\$1$ with probability $\frac{1}{2}$ in a bet. Each bet is independent. Determine ...
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0answers
36 views

Show convergence in distribution of gamma distribution by the central limit theorem

So, I'm not sure how to solve this problem: $X \in \Gamma(a,b)$. Show that $$\frac{X-E[X]}{\sqrt{Var(X)}} \rightarrow^{d} N(0,1)$$ as $a \rightarrow \infty$ Use the central limit theorem. I've come ...
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1answer
37 views

The smallest integer $n$ for a Poisson distribution

Along a stretch of motorway, breakdowns require the summoning of the breakdown services occur with a frequency of 2.4 per day, on average. Assume the breakdowns occur randomly and that they follow ...
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0answers
136 views

Does clever noise exist?

This question is about a random noise, which is called "clever" if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
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1answer
34 views

Tail sigma field generated by i.i.d. sums

The random variables $(X_n)$ are i.i.d. real valued and in $L^1$. Let $S_n := \sum\limits_{i=1}^n X_i$ and $$\mathscr{G_n} := \sigma(S_n,S_{n+1},...).$$ Clearly the sigma fields $\mathscr{G_n}$ are ...
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0answers
94 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = ...
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0answers
24 views

Intersection of Probability measures

Let $X$ be a metric space with Borel $\sigma$-algebra $B(X)$ and suppose that $\mathbb{P}_1$ and $\mathbb{P_2}$ are two probability measures on $(X,B(X))$. Question: Suppose $G$ is an open set ...
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17 views

How to measure the similarity or divergence of two distributions with different supports?

Suppose $X$ and $Y$ are two random variables with the distributions $F_X$ and $F_Y$ on the same support $\Theta$.Then KL divergence $D_{KL}(X||Y)$ is a way to measure the statistical distance between ...
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1answer
30 views

Simple question about weak law of large number with characteristic function version

I was reading a textbook about showing the following Weak Law of Large Number but I stuck in some intermediate steps. Here is the statement I work with Let $\{X_i\}$ be i.i.d. random variables ...
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2answers
18 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
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1answer
11 views

Find distribution of a bernoulli funtion of a unifrom random variable?

I have a uniform random variable $\theta \in [-\pi,+\pi]$. I also have a bernoulli function of this random variable $G(\theta)$, defined as follows, \begin{align} \begin{cases} 1 & \text{if $ - ...
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1answer
17 views

Show that if $X(\omega) = \infty$ then $EX = \infty$

I am trying to show that if $X(\omega) = \infty \space\forall \omega \in A$, $P(A) > 0$ and $X \ge 0$ then $EX = \infty$. The problem comes with a hint: $$EX = E\{X[I(A) + I(A^c)]\} = E[XI(A)] ...
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3answers
135 views

Probability of the horse winning, given the chance of rain

Here's the question: In the past two racing seasons Seahorse has won 55% of the time if the track is dry. On rainy days when the track is muddy he won only 30% of the time. For the next ...
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1answer
39 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ ...
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1answer
31 views

Dirichlet distribution, sum of Beta distributions

I currently have a problem about Dirichlet distributed Variables. In one of the papers I am currently reading it says: Let $S=(S_1,...,S_m)\sim Dir(\delta\omega_1,..., \delta \omega_m)$, with ...
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0answers
48 views

Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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30 views

Probability notation question: differences between undergraduate and graduate texts

Suppose $X$ is a random variable. In most undergraduate math texts, one writes the expected value of $X$ as $\text{E}X$ or $\text{E}[X]$. Similarly, the probability that $X$ is greater than some value ...
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2answers
213 views

How can I justify this set manipulation to show a result in probability?

I am working on the following problem: Proof: Let $a, b, c, d \in \mathbb{R}$ with $a < b$ and $c < d$. We have \begin{align*}P(a < x \leq b, c < Y \leq d) &= P[\{a < x \leq ...
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0answers
16 views

Finding $a$ such that a product of iid $U(0,a)$ converges to $0$ a.s.

Let $a > 0$, let $X_n$, $n \geq 1$, be iid random variables that are uniform on $(0,a)$, and let $Y_n = \prod_{k=1}^n X_k$. Determine, with a proof, all values of $a$ for which $\lim_{n \rightarrow ...
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1answer
25 views

Showing a sequence of Random Variables (Submartingale) Converges.

If $X_n$ and $Y_n$ are non-negative, integrable, and measurable with respect to $\mathscr{F_n}$, with $\mathscr{F_n} \subset \mathscr{F_{n+1}}$, and suppose that $E[X_{n+1}|\mathscr{F_n}] \leq X_n + ...
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19 views

Continuity of Mutual Information

Let $(X,Y) \sim P_{X,Y}$ and $(X',Y') \sim Q_{X,Y}$. Suppose $P_{X,Y}$ and $Q_{X,Y}$ both have the same support and we know, \begin{equation*} D(P \Vert Q) < \epsilon \end{equation*} (where ...
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32 views

conditional expectation of two independent normal random variable

Show that conditional expectation of two independent normal random variable $(X,Y) \approx N(m=(m_1,m_2), \Sigma)$ is equal: $$E[X|Y] = m_1 + \rho \frac{\sigma_X}{\sigma_Y}(Y-m_2)$$ Is there any way ...
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20 views

Reference: proof of Cramer-Rao

I'm looking for a detailed reference of dealing with the proof of the multivariate case of Cramer-Rao lower bound.
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15 views

Expectation of truncated negative binomial distribution

This is what I have done, I would like to know if there is a way to simplify it, solve summations,...
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1answer
33 views

sums and distance of uniform distributions

Let $X$ and $Y$ be two uniformly distributed, independent random variables on the interval $[0,b]$. Let $S = X+Y$ be their sum and $D = |X-Y|$ be their distance. I have a few questions: a) To ...
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1answer
18 views

For X, Y bounded random variables, $E[X^m Y^n] = E[X^m]E[Y^n]$ for all $m, n \geq 0$ implies X, Y are independent

Assume that X, Y are two bounded random variables. If for any integers $m, n \geq 0$, $E[X^m Y^n] = E[X^m]E[Y^n]$, then X and Y are independent. I've worked out that, for sure, if $E[f(X)g(Y)] = ...
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3answers
36 views

Calculate the PMF, mean and variance of X for x=-1,1

An Urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X=1 if a red ball is drawn, and let X=-1 if a white ball is drawn. Give the pmf, mean, and Variance of X. I know ...
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2answers
23 views

Proving the bound $P(|X| \geq x) \leq (M_X(c) + M_X(-c)) e^{-cx}$ under certain conditions

If the moment generating function of a random variable X, $M_X (\lambda) = E[e^{\lambda X}]$, is defined for $|\lambda| < \delta$ with some $\delta > 0$, then $P(|X| \geq x) \leq (M_X(c) + ...
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1answer
36 views

Proving $0.1 ≤ P(AB) ≤ 0.4$ while knowing $P(A)$ and $P(B)$, but not making any further assumptions.

The problem I'm stuck on is: Assume that $P(A) = 0.4$ and $P(B) = 0.7$. Making no further assumptions on $A$ and $B$, prove that $P(AB)$ satisfies $0.1 ≤ P(AB) ≤ 0.4$. This would be easy if we ...