Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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475 views

Domain of a random variable - sample space or probability space?

In most probability theory texts you'll see a random variable definition that goes something like this: A probability space is a triple $(\Omega, \mathcal{F}, P)$, a real valued random variable is a ...
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0answers
31 views

Probability kernel calculations

Suppose $\lambda \Phi(y,\cdot) = \delta_y$ and $\nu_t = \lambda\mu_t \Phi$ satisfies $\lambda \mu_t = \nu_t \lambda$ for each $t\geq 0$. Define $g: S\to \mathbb{R}_{++}$ as $$e^{-\alpha t} \int_S ...
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1answer
58 views

Does reducing 512-bit blocks to 128-bit hashes lead to 1/4 chance of collision?

This is a quote from a cryptography book called Implementing SSL / TLS Using Cryptography and PKI By Joshua Davies. MD5 operates on 512-bit(64 byte) blocks of input. Each block is reduced to a ...
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0answers
104 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon ...
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3answers
54 views

Expectation expression

What is the difference between $E[X^2]$ and $E^2[X]$? Consider $X$ is the number of heads when a fair coin is tossed twice. Please explain clearly in general and w.r.t this example also.
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1answer
50 views

Symmetrisation of function

Consider the probability space $\Omega = \{-1, 0, 1\}$ with the $\sigma$-algebra of all possible events and a probability measure $P$. Consider also the smaller $\sigma$-algebra $$F = \{\emptyset, ...
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0answers
38 views

The mean of $\mu_{P}(\theta)=\frac{1}{Z}P(x|\theta)$

Consider a parametrized probability measure $P(x|\theta)$, that is for each $\theta\in[a,b]$ it is a valid probability measure on $x$. Denote $f(\theta)$ its mean and $\Sigma(\theta)$ its variance. ...
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0answers
73 views

A nice sequence of random variables

Let $f:U\mapsto \mathbb{R}^k$ with $U\subset \mathbb{R}$ be a smooth injective function. Suppose that $\sqrt n(Y_n- Y)\to N(0,\Omega)$ in distribution with $Y=f(X)$. Define $X_n$ by ...
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3answers
79 views

A basic question on integration [closed]

$x^{k}{\rm e}^{-x^{2}}$ decreases to zero "exponentially" when $x \to \pm \infty$, $\int_{\mathbb R}{\rm f}\left(x\right)\,{\rm d}x < \infty$. Which theorem is being used here ?
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2answers
63 views

Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution

I'm trying to prove the statement made by Did in the comments: Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution. So we need to prove that $$\forall t>0: ...
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1answer
39 views

Prove that $P(Y \ge 0)=1$

Let $G$ be a sigma algebra. $Y$ is $G$ measurable and E|Y| < $\infty$ . We also have that $E[ZY] \ge 0$ for all $Z$ such that $Z$ is a bounded, $G$ measurable random variable. Prove that $P(Y \ge ...
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1answer
66 views

problems with probability kernels

Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces and consider a measurable function $\phi: S\to T$. Define a probability kernel $\Phi$ from $S$ to $T$ by $\Phi(x,\cdot) = ...
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0answers
44 views

Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
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0answers
97 views

Kolmogorov extension-type result

I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let $(\Omega, \mathscr{F}, P)$ denote our ...
3
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3answers
287 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
2
votes
1answer
214 views

Draw two cards what is the probability the second is higher than the first? Is my approach correct?

I've seen similar questions posted here before but I was wondering if my method/answer was correct My reasoning was let's say you draw a 2 as your first. Card there are 12 cards with higher values, ...
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0answers
64 views

Criterion of independent random vectors

I would like to know weather the following statements are true or false. Given two random vectors $\boldsymbol{X} = (X_1,\cdots,X_m)$ and $\boldsymbol{Y} = (Y_1,\cdots,Y_n)$, $X_i$ is independent ...
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1answer
44 views

Probability distribution “similar” to Gaussian.

Does there exist a distribution A other than Gaussian such that: 1) linear combination of random variable from A is distribution A 2) easy to integrate, for example find entropy Thank you
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1answer
50 views

Two players $A,B$ throw two dice…

Two players $A,B$ throw two dice. A throw first, and they throw it in turns (i.e. $A,B,A,B,A...$). If $A$ gets sum of $10$ at the dice he wins, if $B$ gets $9$ - he wins. What is the probability ...
1
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1answer
77 views

Showing that a Bessel-squared process is time homogeneous

I'm trying to show that a squared Bessel process is time homogeneous. So far, I've shown that $Y_t=f(X_t)$ is a time-homogeneous Markov process with transition semigroup $(\nu_t)_{t\geq 0}$ w.r.t. ...
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1answer
81 views

Proving a chain is aperiodic, and finding a stationary distribution.

We have an irreducible Markov chain with a not necessarily finite state space. It has a transition matrix $P$ such that $P^2=P$. Prove (1) the chain is aperiodic, and (2) prove $p_{ij}=p_{jj}$ ...
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1answer
44 views

Please explain this conditional expectation equality

I understand that E[X|Y] is a random variable. But I am kind of confused about the following : $$ \int_{\{Y=y_i\}} E[X|Y] dP = E[X|Y=y_i]P(Y=y_i) $$ In the above, P is a probability measure , then ...
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1answer
48 views

Question involving an invariant measure on a Markov chain

Suppose $\mu$ is an invariant measure for a Markov chain with state space $S$ with $\mu(i)p_{ij}=\mu(j)p_{ji}$ $\forall i,j \in S$. Describe a Markov chain with this property. Also, show that $\mu$ is ...
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1answer
31 views

choosing the function of a random variable with the lowest variance w.r.t.o the mean of that random variable?

Consider a real gaussian random variable with mean $\theta$ and unit variance. Let $y$ be an observation of the random variable. The objective is to estimate $\theta$ over all possible $y$. Let ...
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4answers
85 views

determining the amount of total questions needed in a game given the probabilty

I'm creating a game and can't seem to quite figure this out - driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn't change. so ...
3
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1answer
107 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
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0answers
96 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
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0answers
25 views

A basic question on distribution function and stieljes integral [duplicate]

Is it true that $$\int_{\Bbb R} G(x)dF(x) + \int_{\Bbb R} F(y)dG(y) = 1$$ where $F$ and $G$ are distribution functions of $X$ and $Y$ such that both have no common jump points i.e. $P(X=Y) = 0$ ? ...
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0answers
48 views

About one stoping time definition in Chung's book (A Course in Probability Theory)

In Chung's book, he defines the stopping time $ \alpha^k $ in the following way. $\alpha^1 = \alpha$; $\alpha^{k+1}(\omega) = \alpha^k(\tau^\alpha \omega)$; where $\tau^\alpha$ is the ...
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2answers
120 views

Is a non-negative random variable with zero mean almost surely zero?

We have proven the following in class: If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements). Does it ...
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0answers
28 views

Conditional probability density function of two variables

Assume I am given a continuous function $f(x_1,\cdots x_n)$ over some $n$-dimensional region $R$. I want to show that this function is a jointly continuous probability density function. May I show ...
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1answer
64 views

How to make use of the hint for proving $\text{CLT} \implies \text{WLLN}$?

I've seen an exercise where one is asked to prove that the central limit theorem implies the weak law of large numbers. The author gave the following hint: "First prove that convergence in ...
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1answer
62 views

A basic question on expectation of distribution composed random variables

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$. If $F$ and $G$ have no common jumps then I need to show that $E[F(Y)] + E[G(X)] = 1$. How to proceed here ? ...
0
votes
1answer
159 views

Invariance of Brownian motion under orthogonal transformations

Let $\left(B_t\right)_{t \in [0,\infty)}$ be an $n$-dimensional Brownian motion with start at $x \in \mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ real matrix. I'm trying to show that $AB$ ...
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0answers
53 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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0answers
36 views

Proving that $ (\mathbb E [X^n])^{1/n}\leq (\mathbb E [X^m])^{1/m}$ for $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$

How to prove that for a positive essentially bounded random variable $X$ satisfing $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$ and for any $m,n \in \mathbb N^*$ with $m\geq n$ we have $$ (\mathbb E ...
0
votes
1answer
148 views

Bayes Theorem Drug Testing

A large company gives a new employee a drug test. The False-Positive rate is 3% and the False-Negative rate is 2%. In addition, 2% of the population use the drug. The employee tests positive ...
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0answers
37 views

Maximum profit from random variables

Let $X_i$ are $N$ random variables and $a_i$ are positive integers where $ -1 \lt X_i \lt 1$. My goal is to maximize the average profit $P= \sum_{i=1}^N a_iX_i$. What is the optimal number $N$ ...
0
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1answer
88 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
2
votes
1answer
81 views

Three questions about ucp convergence

We say that a sequence of processes $X^n$ converges to a process $X$ uniformly on compacts in probability if for all $\epsilon >0, t>0$ $$P[\sup_{s\le t}|X^n_s-X_s|>\epsilon]\to 0 $$ for ...
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1answer
44 views

Tossing Dice repeatedly, probability that 2nd trial had more tosses than 1st one

John repeatedly tosses a die until a six occurs for the first time. Alice then repeats the experiment. What is the probability that Alice made more tosses than John?
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2answers
100 views

About Example 1.8.2 in Durrett: Probability Theory and Examples

The example is about tail $\sigma$-field. Given i.i.d. r.v. $ X_1, X_2, \dots $ and the partial sum $ S_n = X_1 + \dots + X_n $. The example says that $\{ \limsup_{n\rightarrow\infty} S_n > 0 \} ...
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1answer
62 views

Measurability from taking out property of conditional expectation

Two random variables $X$ and $Y$, and a sub $\sigma$-algebra $\mathcal G$. Does $\mathrm E(YX) = \mathrm E(Y \, \mathrm E(X|\mathcal G))$ imply $Y$ is $\mathcal G$-measurable? Do we need to require ...
3
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1answer
94 views

Topology of weak convergence

Edited: Thanks to etienne. I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak ...
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0answers
50 views

The subspace which $X-E(X|\mathcal G)$ is orthogonal to, and the set of r.v.s generating the same $\mathcal G$.

$X$ is a random variable. I wonder if there are some or no relations between the subspace which $X - E(X|\mathcal G)$ is orthogonal to, which is the set of all random variables which are both ...
2
votes
1answer
78 views

The probability of a Brownian motion's tail event is unaffected by the starting point

Consider the measurable space $\left(\mathbf{C}\left[0,\infty\right), \mathcal{B}\left(\mathbf{C}\left[0,\infty\right)\right)\right)$ and the stochastic process $\left(X_t\right)_{t \in ...
2
votes
3answers
74 views

Proof expectation of bernoulli distribution

Suppose we have: $P(X=k) = (1-p)^k p$ $$E(X) = \sum^{\infty}_{k=0} kP(X=k)= \sum^{\infty}_{k=0} kp(1-p)^k = p(1-p) \frac{1}{p^2}=\frac{1-p}{p}$$ What I do not get is the step in the equation ...
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1answer
69 views

Conditional expectation of a Poisson random variable: confusing sums

In Probabilty and Random Processes by Grimmett and Stirzaker, the following example is given (page 68): My question is: how are the last three (math) lines true? Specifically, how do these two ...
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0answers
27 views

Can we prove this after removing one condition?

mathematicians! I want to ask about redundant condition in the below proof. Suppose there exists a partition $A_{0}, A_{1}, \ldots, A_{k}$ of $\chi$ such that $P(X \in A_{0}) = 0$ and $f_{X}(x)$ is ...
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1answer
27 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...