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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3
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77 views

how to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale.and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
0
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1answer
33 views

Definition of the probability distribution of a random variable?

I have a vocabulary problem. I understand that a "probability distribution" is a function from a sigma algebra to the reals. But then what is a "probability distribution" in "the probability ...
0
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3answers
73 views

Expectation of a squared random variable and of its absolute value

Is it true that if $\mathbb{E}[X^2]<\infty$ then $\mathbb{E}[|X|]<\infty$? If so, why?
0
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1answer
146 views

Convergence of events in a probability space with respect to $L^2$

Define for events $X, Y$ that $d(X,Y) = P((X-Y) \cup (Y-X)) $ = $ P(X \bigtriangleup Y) $, show that $d(X_n,X) \rightarrow 0$ if and only if $\chi_{X_n}$ converges in $L^2$ to $\chi_X$ (these are ...
0
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1answer
219 views

What is the probability of picking Exactly 1 red marble and than not 1 red marble? without rep.

A urn has 3 red marbles, 2 blue marbles, 1white, 1 black 1 brown. What is the probability of getting exactly 1 red marble than not 1 red marble? What is the probability of getting at least 1 red ...
2
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1answer
121 views

Is it possible to flip tails indefinitely?

If someone makes the argument that it is impossible to flip tails $n$ times on a two-sided coin, then we can argue there is a $1$ in $2^n$ chance. There is not a definable point at which it becomes ...
1
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1answer
25 views

Moment Generating Function of a Positive Standard Normal

I'm trying to calculate the m.g.f. of the positive standard normal random variable X, by which the p.d.f. is $f_X(x) = \sqrt{\frac{2}{\pi}}e^{-x^2/2}$ valid for $x \geq 0$. So: $$E(e^{tX}) = ...
0
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2answers
56 views

Counter example for Joint Density function expression

For this expression: $$\int_{x=-\infty}^{\infty}f(x,x)dx\leqslant 1$$ I know that the equal signs is the only thing that happens, not the less than sign. By using the condition to be joint density ...
0
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0answers
30 views

Correlation: How to extend it from pairs to further random variables?

How can one determine the correlation coefficients (or their intervals) between $n$ standard-normal random variables $X_i$, $i=1,...,n$, when $X_i$ correlates with $X_{i+1}$ with correlation ...
1
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0answers
45 views

Monotonicity of a probability sequence

Let $\mathbb{P}$ be a probability measure on $\mathbb{R}^n$, so $\mathbb{P}( \mathbb{R}^n ) = 1$. Consider a mapping $f: \mathcal{P}( \mathbb{R}^n ) \rightarrow \{ 0, 1, ..., F \}$, where ...
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0answers
44 views

Bernoulli process with different parameters

Let $X_n$ be independent random variables taking values $0$ and $1$ with $P(X=1) = p_i$. 1) Show the strong law of large numbers. 2) Show the central limit theorem holds iff $\sum p_i(1-p_i) = ...
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0answers
45 views

Poisson Decomposition Problem & Interarrivals

My Attempt: As for (c) and (d) I really do not know how to start but I sense it has to do with interarrival times. Thanks !
0
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1answer
19 views

Distribution of outgoing calls

I know that stream of independent events obeys Poisson distribution (so summary stream of calls that achieved Phone Station (PBX) will obey Poisson distribution). But I don't think separate streams ...
1
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1answer
86 views

Stopping times of Markov chains

I have the following problem: Consider a state space $E$ and a Markov chain $X$ on $E$ with transition matrix $Q$ such that for every $x \in E$, $Q(x,x)<1$. Define: $\tau:=\inf\{n\geq 1:X_n\neq ...
1
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1answer
134 views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...
2
votes
1answer
770 views

Prove that Cov(X,Y)=Cov(X,E[Y|X])

I've been working on this problem for 3 hours now, and my complete lack of progress is getting disheartening. I've looked up definitions, proofs, and have even seen a solution for this particular ...
2
votes
2answers
110 views

I want to show $E(X)=\int_{0}^{\infty}P(X\ge x)dx$ for non-negative random variable $X$

Show that for a non-negative random variable $X$, $$\mathbb E(X)=\int_{0}^{\infty}\mathbb P(X\ge x)dx.$$ I started with $$\mathbb ...
3
votes
1answer
95 views

Functions and convergence in law

Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ ...
2
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0answers
55 views

Understanding a theorem from “Probability theory of Banach Spaces ” book.

I don't understand the proof after "The hypothesis of the theorem indicate...... , can someone kindly explain it for me . Thanks :)
0
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0answers
65 views

Sampling uniformly from n-sphere using spherical coordinates

It has been explained here why sampling from n-sphere is not achievable with naive parametrization. And it explains how to correct it for 3 dimensions. Can somebody please guide me what is the correct ...
2
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1answer
51 views

Convergence in probability inverse of random variable

if $ X_n \to 1 $ in probability i need to prove that $X_n^{-1} \to 1$ under probability. I got till the point that i need to prove the following probabilities 0, but don't know how to prove them? i.e ...
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1answer
51 views

The intergral $I=\int _0^{\beta }f(x)dx$ is given,for $\alpha,\beta \in \mathbb{Z}$ ,how can we find $\int_0^{\alpha\beta}f(x)dx$ in terms of $I$

i am working with a gaussian normal distribution function in probability,i am given values for the integral when $z\le 4$ and i want to find a value $z=8$,in general if $z=4$ is given , how to find ...
1
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1answer
63 views

weaker version of the martingale convergence theorem

Let $\mathcal{A}_n$ be a sequence of finite sigma-algebras, let $\mathcal{B}_{q,p}= \sigma(\mathcal{A}_n, q \geq n \geq p )$. Moreover, we suppose $\mathcal{A}_k \subset \mathcal{B}_{\infty,p}$ for ...
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0answers
63 views

Joint probability distributions of variables satisfying a certain constraints

Here is my question, given a set of random variables - {x_i}, i=1,2, ...n. And the corresponding pdfs are given by {PDF_i}, i=1,2, ...n. Now if I were it impose a certain set of constraints on ...
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1answer
33 views

combinatorics -probability of burned electric bulbs in row

Given 50 bulbs in a row.In a given time (not maintenance problem) the probability of a burned bulb is 0.1 . Please calculate the probability that the last 5 bulbs in row are burned. I really doubt ...
2
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2answers
1k views

How to calculate probabilities for football match?

I am looking for the acceptible methods to calculate probability chances "1 x 2" on football matches. I read a lot of articles, but there are no useful information that can help me. I started to ...
0
votes
0answers
41 views

I want to show (i) and (ii) are equivalent.

Show that the following are equivalent: (i) A Family $A_{i}$ of events is independent; (ii) The family $\sigma(\mathbb 1_{A_{i}})$ of $\sigma$-algebra is independent. thanks for help.
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1answer
40 views

Finding the general pattern form

Suppose a Markov chain given as follows. $p_{ii}=1-3a$ And $p_{ij}=a\ \forall i\ne j$ where $P=(p_{ij}), 1\le i,j\le 4$. Find $P_{1,1}^n$. Attempts: I have tried to compute for the case $n=1, 2, 3$. ...
0
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1answer
46 views

Finding Calculus or Probability error in basic continuous probability problem

I am having a lot of difficulty spotting my error in the following probability problem. The joint probability density function of $X$ and $Y$ is given by $f(x,y) = c\left ...
3
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0answers
31 views

Normal random variables, joint distribution

If I have two normally distributed random variables, is their joint distribution always elliptical, i.e. fully characterized by a mean vector and variance-covariance matrix?
0
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1answer
51 views

Random Process not mean square continuous

Does a WSS (Wide Sense Stationary) process exist which is not mean square continuous? If so, can you give me an example. Note: A WSS process is mean square continuous iff the autocorrelation ...
0
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1answer
34 views

Show $\sum_{y\ \in\ Y\ \ } \sum_{x\ \in \ X\ :\ x \le \ y} \left( \left(\frac{1}{6-x+1} \right) \left(\frac{2(6-x+1)-1}{36} \right) \right) = 1$

Two fair dice are rolled. Find the joint probability mass function of $X$ and $Y$ when $X$ is the smallest and $Y$ is the largest value obtained on the dice. Reasoning this out (but leaving ...
3
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0answers
38 views

Show equivalences concerning independence

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. We say, that $(E_i\in\mathcal{A}:i\in I)$ is a family of independent events, if for any finite subset $I_0\subset I$ it is ...
0
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1answer
49 views

0-1 Law: Applications?

The question is more open to a debate rather than a mathematical explanation: If $(A_n)_{n\in \mathbb{N}}$ is a sequence of $\sigma$-algebras. $\mathfrak{A}_n := \sigma(\bigcup_{m\ge n} A_m)$ is the ...
4
votes
1answer
62 views

Conditional expectation by $\sigma (G_n,Y)$ when $Y$ is $G_\infty$-measurable

Let $G_n$ be a filtration (an increasing sequence of sigma-algebras), $Y$ a random variable that is $G_\infty$-measurable, and $X$ a random variable. Is it true that in $L^2$-norm, $$ \mathbb{E}[X| ...
3
votes
1answer
52 views

the density of the sum of $n$ random variables with uniform distribution on $(-1,1)$

Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density $$f(x)= \frac{1}{\pi} ...
0
votes
1answer
354 views

Find unknown value in probability density function

Suppose that a random variable Y has a probability density function given by f (y) = ky^3*e^(-y/2), y > 0, and 0, elsewhere. Find the value of k that makes f (y) a density function. I found that k=1 ...
1
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2answers
210 views

Expected value and variance on exponential distribution

The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y^2 relate the cost C of ...
0
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1answer
52 views

is any upper bound for mean square error of an unbiased estimator?

There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of ...
2
votes
1answer
154 views

Conditional Expectation of martingale at stopping time

I am trying to understand the implications of the optimal stopping theorem, which is why I tought of the following problem. Consider the continuous-time Martingale $X = (X_t)_{t \geq 0}$ and the ...
1
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0answers
88 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
3
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0answers
33 views

Number of times above a linear boundary for a finite variance random walk

I consider a random walk $(S_n)$ with mean zero and finite variance, and $\epsilon>0$. Is it true that $$ \mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ? $$ ...
0
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1answer
200 views

Distribution function (CDF) of the sum of two random variables + law of iterated expectations

I'm taking my first probability class, and we're studying sums of independent random variables. We're using Ross's First Course in Probability. It states the definition of a convolution, but doesn't ...
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0answers
128 views

Analogy of $[0,1]^n$

A probability maps a set (such as 1 dimensional path) to $[0,1]$. Is there a sufficient and necessary analogy of $[0,1]^n$ to a $n-$dimensional cube ? or to $n-$dimensional other well known objects ? ...
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0answers
70 views

What method to use for this(poison binomial or binomial distribution?

The experiment E is to take a required computer science course. The outcomes are the grades (A, B+,B, C+, C, D, F). Suppose the experiment is repeated 5 times, once for each of CS111, CS112,CS113, ...
6
votes
1answer
73 views

Measurability of the pushforward operator on measures

Let $X$, $Y$ and $Y'$ be (standard) Borel spaces. We let $\mathcal B(X)$ be the Borel $\sigma$-algebra of $X$ and $\mathcal P(X)$ to be the space of all Borel probability distributions on $X$ endowed ...
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votes
1answer
44 views

Ergodic, stationary Probability Vector [closed]

Hello, I really need help with this question. I have no idea how to do it.
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0answers
61 views

random variable probability(binary tree)

I have this problem on random variables and a binary tree. The problem goes like this: The experiment is creating a binary tree with 4 nodes. The random variable X is the number of leaves. My first ...
3
votes
1answer
56 views

Does convergence of $\lim_{n \rightarrow \infty} \int h d\mu_n$ for all continuous, bounded $h$ imply weak convergence of $(\mu_n)$?

Let $(\mu_n)$ be a sequence of positive finite Borel measures on $\mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} \int h d\mu_n$ converges for every bounded, continuous function $h$ on $\mathbb{R}$. ...
4
votes
1answer
209 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...