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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28 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, ...
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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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60 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 ...
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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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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? ...
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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. ...
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1answer
24 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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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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24 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}} = ...
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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?
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33 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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29 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?
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1answer
32 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 ...
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1answer
25 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 ...
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1answer
49 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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21 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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37 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 ...
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27 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 ...
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1answer
45 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
33 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 ...
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1answer
31 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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23 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 ...
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1answer
24 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 ...
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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 ...
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46 views

Vector distribution after Girsanov transform

Let $X$ be a gaussian vector under $P$ and $U$ a variable such that the vector $(X,U)$ is gaussian. $dQ = Y dP$ with $Y = e^{(U −E_p(U) − 1/2 var_p[U])}$. I have to show that $X$ is gaussian under ...
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20 views

Tree-width of a graph

What is the tree width of the graph? Here are the relevant definitions from my textbook: We define the width of an induced graph to be the number of nodes in the largest clique in the graph ...
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1answer
37 views

Regular dependence

I have seen the definition of "regular dependence" in many books (usually old books), but I could not fully understand that definition, hope you can help me understand it. The dependence of $X$ and ...
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1answer
46 views

Why do we need to use random variables

In my statistics textbook (The Practice of Statistics by Starnes, Yates, and Moore) an example is given. In it, 21 students are each given three glasses of water. Two are filled with tap water and one ...
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27 views

Itô Excursion Measure

I am looking for any source of information regarding Itô Excursion Measure (for Brownian Motion). I am looking for a selfcointained reference (Though I have basic knowledge on Local Times and Poisson ...
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2answers
58 views

Probability distribution of $\min(X,Y)$ given that $\max(X,Y)>1/2$ [closed]

Suppose $X$ and $Y$ are two independent random variables. What is the value of $\Pr[\min(X,Y) \leq z \mid \max(X,Y) >1/2]$? They both follow a Uniform distribution with parameters 0 and 1
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1answer
30 views

Mean value theorem inside the Expectation

Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$ \operatorname{E} \left[ \int_0^t X_s \, ...
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30 views

Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
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32 views

Frequency in sample path of Markov Chain

Consider a two state Markov chain with states $S=\{l,H\}$. Suppose the probability of transitioning from $L$ to $H$ is $\alpha$ and the probability of transitioning from $H$ to $H$ is $\beta$, where ...
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1answer
18 views

When a r.v. admits mean and variance?

If $X$ is a r.v. on $(\Omega,\mathcal E, P)$ I have $E(X)=\int_\Omega X \, dP$ $\mathrm{Var}(X)=\int_\Omega (X-E(X))^2 \, dP=\int_\Omega X^2 \, dP - 2E(X) \int_\Omega X \, dP + E(X)^2 $ so $X ...
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1answer
28 views

Question on chi square transformation

I'm trying to follow the maths of a population genetics paper. Basically, I am stuck in a manipulation where the authors notice that a given function has the shape of a chi square distribution and ...
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1answer
18 views

Markov property for a stochastic process with discrete state space.

Consider a stochastic process $\{X_s\}_{s\in\mathcal S\subseteq\mathbb R}$ with value in $(\mathbb R,\mathcal B(\mathbb R))$ adapted to a filtration $\{\mathcal F_s\}$ (we can suppose that ...
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1answer
54 views

Suitable martingales and optional stopping theorem

Starting at value 0, the fortune of an investor increases per week by 200 with probability 3/8, remains constant with probability 3/8 and decreases by 200 with probability 2/8. The weekly increments ...
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63 views

If $ P(A) = 0 $ is $ A $ a null event?

I know that $ P(\text{null event}) = 0 $, but is the reverse true? i.e. if $ P(A) = 0 $ is $ A $ a null event? I'm not too sure I even understand what a null event is, to be honest. Could anyone give ...
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1answer
14 views

Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...
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40 views

Formal Proof: P(A∩B'∩C') = P(A) - P(A∩B) - P(A∩C) + P(A∩B∩C)

I'm trying to prove the following: $\newcommand{\P}{\operatorname{\bf P}}\P(A\cap \overline{B}\cap\overline{C}) = \P(A) - \P(A\cap B) - \P(A\cap C) + \P(A\cap B\cap C)$ I can explain it with a venn ...
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1answer
40 views

If $f$ is a pdf can we construct $g$ such that $x\sim U[0,1)$ implies $g(x)\sim f$

Let $f$ be some pdf over $[0,1)$. Here is my question: does there always exist an infinite partition $\{X_{s}\}_{s\,\in\, \mathrm{support}(f)}$ of $[0,1)$ such that if we define $g(x):[0,1)\rightarrow ...
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1answer
26 views

Why is $\limsup\limits_{n\to\infty}X_n$, $C_{\infty}$-measurable?

Why is $\limsup\limits_{n\to\infty}X_n$, $C_{\infty}$-measurable ? If $\mathcal B_n=\sigma(X_n)$,$\quad$$\mathcal C_n=\sigma\left(\bigcup_{m\ge n}\mathcal B_n\right)$,$\quad$$\mathcal ...
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27 views

Markov chain with Poisson distribution

Let $X \in \mathbb{R}^+$ and $Y \in \mathbb{Z}^+$ be the Random Variables (RVs) where the condition PDF $f_{Y|X}(y|x)$ follows a Poisson distribution as $$ f_{Y|X}(y|x) = ...
4
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1answer
142 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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44 views

Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...
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42 views

Simplifying an integral involving Gaussian PDF

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty ...
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62 views

rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
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1answer
38 views

Expected value and covariance of compound Poisson process

$Y_1,Y_2,...$ are independent random variables with a distribution identical to that of $Y$. $N(t)$ is a poisson process with parameter $\lambda$. $$X(t)=\sum\limits_{n=1}^{N(t)}Y_n$$ Find the ...
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1answer
33 views

Is $Q_n(A_n)=E(L_n A_n)$ a probability measure?

Let $(\Omega, F, F_t, P)$ be a filtered probability space and $(L_n)_{n \geq0}$ a family of positive and $F_t$ adapted random variables. I have to find the conditions for which $Q_n$, defined on ...
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2answers
44 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...