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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2
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1answer
59 views

Proving that three events are mutually independent

Suppose that: the events $A$ and $B\cap C$ are independent. the events $B$ and $A\cap C$ are independent. the events $C$ and $A\cap B$ are independent. the events $A$ and $B\cup C$ ...
0
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1answer
19 views

Probability densities and Absolute continuity

I've not deep knowledge in measure theory/real analysis but just few concepts given me during this second year probability course. I'm trying by myself to understand more, but I don't want to dive in ...
4
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2answers
110 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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0answers
33 views

Finding an example where the sum of iid rv's is infinite almost surely

$Y_1, Y_2, \dots$ are iid non-negative random variables. Let $S = \sum_{n=1}^\infty \alpha^n Y_n$, where $\alpha \in (0,1)$. Now, $EY < \infty$ implies that $S < \infty$ almost surely. Can ...
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0answers
23 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
1
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1answer
18 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0
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1answer
30 views

Simple conditional probability inequality

I'm reading on some branching process theory in Harris' Theory of Branching Processes and encountered an inequality which looks simple but is eluding me. The full version is a bit complicated to ...
1
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1answer
24 views

Probability Game Question

I am new to probability. I am trying to solve the following problem. In a game, probability of winning the game is $w$ & losing the game is $l$ & probability of game continuing is $(1 - w - ...
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0answers
40 views

NUMBER OF ATOMS IN A SIGMA-ALGEBRA [on hold]

I have been trying to solve the followIng question. DESCRIBE THE SMALLEST SIGMA ALGEBRA CONTAINING 'n' ARBITRARY SUBSETS OF THE SAMPLE SPACE.GIVE AN UPPER BOUND FOR THE NUMBER OF SETS IN THIS SIGMA ...
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0answers
34 views

Chernoff Binomial Bound

I am reading a paper and the following Chernoff-type bound is presented: For X~Bin(n,p) and a>0, the following bounds for lower and upper tail, respectively, hold: $$\Pr[X\le np-a]\le ...
2
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3answers
40 views

Finding the expected value of a function of random variables

I'm having troubles with finding marginal density functions and expected values in my probability theory class. I was hoping someone would be able to walk me through the solution to this question (I ...
0
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0answers
29 views

Geometric series: Convergence under which conditions?

For which functions $p(n)$ does $$\sum_{i=0}^{n} i (p(n))^i \rightarrow \infty$$ but $$ \frac{1}{n} \sum_{i=0}^{n} i (p(n))^i \rightarrow 0$$ Or stated differently: I want $$ 1 \ll \sum_{i=0}^{n} i ...
1
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1answer
40 views

Poisson, Gamma distribution example.

Can someone explain me answer for these questions? Suppose customers arrive at a store as a Poisson process with λ = 10 customers per hour. The Poisson process of X ∼ Poisson(λ) the time until k ...
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0answers
41 views

When can one represent the conditional expectation $E[X|Y]$ as $g(Y)$ with continuous $g$?

Given two random variables $X$ and $Y$ we know that $E[X|Y] = g(Y)$ where $g$ is a Borel function. Is it a good question to ask under which condition there exists a function $g$ which will be ...
3
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1answer
40 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
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0answers
44 views

Is my interpretation of Bayesian probability and inference correct?

I have the following interpretation of the Bayesian probability and inference (without referring to Measure Theory, I am still at the very beginning of learning it): Let's say we have five random ...
-3
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1answer
31 views

Derive/ prove: p(a,b|c) = p(a|b,c).p(b|c)

How can this expression be derived? p(a,b|c) = p(a|b,c).p(b|c) where a,b,c are random variables. UPDATE: from the following ...
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0answers
16 views

Category theoretic view of coupling measures/RVs

Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, ...
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0answers
23 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
2
votes
1answer
33 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
0
votes
2answers
33 views

Finding independence of two random variables

We're learning about independent random variables in the context of multivariate probability distributions and I just need some help with this one question. If $f(y_1, y_2)=6y_1^2y_2$ when $0\leq y_1 ...
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1answer
38 views

Proof of “continuity from above” and “continuity from below” from the axioms of probability

One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them. if ...
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0answers
20 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
2
votes
1answer
46 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
0
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0answers
34 views

Uniform intergrability of the maximum of a series of iid random variables [on hold]

Suppose $X_n$ are independent and identically distributed $L_1$ random variables. Let $T_n = \max_{1\leq j\leq n} X_n$. Is $\frac{T_n}{n}$ uniformly integrable?
0
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1answer
25 views

Asymptotic Equivalence implies same asymptotic distribution?

A book I'm reading stated that if we have nonnegative random variables, and if $X_n\to X > 0$ in distribution and $\frac{Y_n}{X_n} \to 1$ in probability then $Y_n \to X$ in distribution. However, ...
0
votes
2answers
25 views

Show that $A \cup B$ and $C$ are independent as well

Show that if 3 events ($A$, $B$ and $C$) are independent, that the events $A \cup B$ and $C$ are independent as well. It seems pretty logically straightforward but how do you show this statistically. ...
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2answers
59 views

Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
-2
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0answers
37 views

Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
0
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0answers
50 views

Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
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2answers
42 views

Please just tell me if my working is correct.

It is known that $P(X = n) = p_n = \frac{2}{3^n}$ $X$ is the number of attempts needed to win the lottery. The question: Find $P(X>5)$ My take: $P(X>5) = 1 - P(X\leq 5)$ => is this correct? ...
0
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1answer
31 views

Show that $k = 2$ (Use the fact that $\sum_{n=1}^{\infty}p_n = 1$)

The set of numbers $p_1, p_2, p_3, ..., p_\infty$ such that $P(X = n) = p_n = \frac{k}{3^n}$ define an infinite probability space associated with the number of attempts, X needed to win the lottery. ...
0
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1answer
23 views

Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...
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0answers
13 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
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0answers
22 views

Intuition about generating functions

I am trying to gain some intuition about moment generating functions. In particular, for a random variable $X$, we have $$ \newcommand{\E}[1]{\mathbf{E}\!\left[#1\right]} M_X(t) = \E{e^{Xt}} = ...
1
vote
1answer
39 views

Why is this statement false?

If $P(A^C) = \alpha$ and $P(B^C)=\beta$ then $P(A \cup B) < 1 - \alpha - \beta$ It is false. I know that and I can visualize it but how can I show it statistically?
0
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2answers
32 views

If $P(A\setminus B)\geq P(B\setminus A)$, then $P(A) \leq P(B):\;$ Why false?

If $P(A\setminus B)\geq P(B\setminus A)$, then $P(A) \leq P(B)$. I know that it's false using common sense but how do I show that statistically?
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0answers
28 views

Probability Distributions and Random Discrete Variables

How do you read this? For (a) do we let $X= 1/6, 1/2, 1/5$ and $2/15$ and sub into the equation, $$ Y=X^2-2X. $$ How do we go about solving this?
2
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1answer
31 views

Difference between $\lim P[…]$ and $P[ \lim ]$

In a Galton-Watson branching process the extinction probability is sometimes given by $$\lim_{t \rightarrow \infty} P[X(t)=0]$$ and sometimes as $$ P[\lim_{t \rightarrow \infty}X(t)=0]$$ Is there a ...
0
votes
1answer
23 views

Conditional Probability using a Matrix

I understand how to find P1: that is simply: P(D1|D0)=0.8 P(W1|D0)=0.2 P(D1|W1)=0.4 P(W1|W0)=0.6 I do not however, understand how to find P2 using the matrix. Normally I would solve it as ...
2
votes
1answer
41 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
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2answers
18 views

Proving the Probability of an Event Through Bayes Theorem.

The question goes as such: An event A can occur if only one of the mutually exclusive events B1, B2, or B3 occur. Show that P(A) = P(B1)P(A|B1)+P(B2)(A|B2)+P(B3)*(A|B3) my working out: P[A|(B1 U B2 ...
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0answers
34 views

Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.) Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal ...
3
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0answers
25 views

Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with ...
0
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1answer
43 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
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2answers
27 views

Finding the mean and variance of an exponential probability distribution

I'm taking a probability theory course, and I'm struggling a bit with gamma and exponential distributions. Here's a question that I'm stuck on: The length of time Y necessary to complete a key ...
0
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1answer
30 views

How to check hypothesis in statistical data?

I have a statistical problem. In a city there are some hostels which differ by the number of rooms. The input data are the following. In a table there is information about hostels and corresponding ...
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0answers
21 views

Secretary problem *without* each ordering equally likely

Also known as: the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. See http://en.wikipedia.org/wiki/Secretary_problem $n/e$ is ...
0
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1answer
23 views

Linear Combinations of Normal Variables and Independence

I'm reviewing for a final and am unsure how to do one of the review questions re: bivariate normals. I'm given: $ Suppose (X,Y) \sim BN(\mu_x=0, \mu_y=0, \sigma_x^2=1, \sigma_y^2=1, \rho=0.6)$ Find ...
0
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1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...