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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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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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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35 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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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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90 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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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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0answers
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
212 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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17 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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30 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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19 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
17 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
35 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
20 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 ...
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1answer
31 views

Problem with Renewal process and holding times.

Let $(N_t)_{t \geq 0}$ be a ordinary renewal process, i.e. the interarrival times of the processes jumps say $(T_k-T_{k-1})_{k\in\mathbb{N}}$ is an i.i.d sequence. In an example in my book on the ...
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1answer
28 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
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1answer
126 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
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1answer
41 views

$P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$ for all $a < b$ implies that $lim_{n \rightarrow \infty} X_n$ exists a.e.

Suppose for $a<b$ we have $P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$. Then $lim_{n \rightarrow \infty} X_n$ exists a.e. but may be infinite. Here "i.o." means "infinitely often"; for any ...
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1answer
117 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
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1answer
24 views

Common distribution of order statistics

Let $X_1,\dots,X_n$ be $iid$ with distribution function F and $Y_1,\dots,Y_n$ $iid$ with distribution function G. I've proved for some function $g$ that $X_1$ is equal in distribution to $g(Y_1)$, ...
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10 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
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1answer
23 views

Can anyone help me find the variance of this expression?

I have a vector of the form \begin{align} {\bf a }= \frac{1}{\sqrt{N}}[1, e^{jA}, e^{j2A},\cdots, e^{j(N-1)A}]^T \end{align} where A and N are constants. I also have a vector N of i.i.d ...
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31 views

Convergence of unbounded functionals

Let $\mu_n$ be a sequence of random measures on the metric space $(\mathbb{R},d)$ that converges in some mode to a measure $\mu$. The definition of weak convergence is: \begin{equation} ...
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3answers
60 views

Why is a probability density function nonnegative?

Let $X$ be a random variable and its density $f$ be defined to be the derivative of its distribution function $F$, i.e. $$\Pr(a< X\le b)=F(b)-F(a)=\int_a^bf(x)\operatorname{dx}$$ Now let ...
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1answer
39 views

Expectation of truncated Poisson Distribution

I have found that if I have a $Y \sim \mathrm{Poi}(\lambda)$ and $Z=Y \mid Y>0$ then I say $$f_Z(k)=g(k)=Pr(Y=k\mid k>0)=\frac{\lambda^k}{k! (e^\lambda-1)}$$ Now I am trying to compute ...
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1answer
13 views

How to find the mean variable of a normal distribution with a given probability and standard deviation?

We have a machine that produces µ g of pasta to be stored in their package, with a standard deviation of 20g. It follows a normal distribution. And we don't want it to produce more than the package's ...
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1answer
20 views

A simple question about the definition of martingales

The definition of Martingale denotes that $E(M_{n+1}|\mathcal{F}_n)=M_n$. This implies $E(E(M_{n+1}|\mathcal{F}_n))=E(M_n)$. Then does it mean that $E(M_{n+1})=E(M_n)$ using the tower property? If so, ...
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10 views

Application of Strong Markov Property

Theorem SMP (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or $\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many values. Then ...
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1answer
35 views

Skorokhod's theorem and summary of convergence of sequence of RVs

This question is about convergence of RVs (when convergence in one sense implies other convergence modes). I would like to have a big picture on convergence modes and various implications between ...
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1answer
48 views

Prove that this sequence converges almost surely

Suppose that $(X_n)_{n\ge1}$ is a sequence of independent random variables with $E[|X_n|] < \infty$ for all $n$ and $E(X_n) = \mu$. Prove that $$\sum_{n=1}^{\infty}\frac{1}{2^n}X_n = \mu \; a.s$$ ...
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2answers
20 views

7 white balls and 3 black balls in a box. You take out in sequence 3 balls at random without replacement. Probability that all 3 balls are black?

What i did was say that the probability the first ball is black is 3/10, then since we removed one the probability the second is black is 2/9 and third is ⅛. So then the probability all three are ...
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2answers
16 views

Marginal P.M.F and Conditional Expectation?

I have a joint density function that is formulated as follows: $$ f_{X,Y}(k,y) = \begin{cases} \frac{\partial{P(X=k, Y\le y)}}{\partial y} = \lambda \frac{(\lambda y)^k}{k!}e^{-2\lambda y} & ...