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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$f \le 1 \Rightarrow f =1 $ a.s.

I know the title doesn't say much, but I hope you'll help me nonetheless. Here's my problem. Let $P, Q$ be two probabilistic measures, $P$ is atomless and the measures have the same independent ...
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Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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24 views

Relations between stochastic Big O-Notation, limsup and lim?

I have to solve a list of exercises on probability theory but I'm having several problems in understanding the following questions; could you help me? Thank you! Let $||.||$ indicate the Euclidean ...
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40 views

Law of large numbers variant?

I have the following: Let $(X_n)$ be a sequence of i.i.d. random variables. (a) Assume $\frac{1}{n} S_n=\frac{1}{n} \sum_{i=1}^n X_i$ converges a.s. to a real-valued random variable $Y$. Show that ...
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32 views

a martingale equality

Let $X_{t}$ a positive continuous martingale satisfying: $\lim_{t\longrightarrow \infty}X_{t}=0 $ ps and $X_{0}=a \in {R_{+}}$ Show that $\textbf{P}(\sup_{t\geq0}X_{t} \geq b)=\frac{a}{b}$ , a < ...
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33 views

Difference Uniform rv's

Let $U_{1}\sim U(0,1)$ be a standard uniform random variable. Is $U_{1}-U_{1}$ uniformly distributed? I've been trying to work this out as follows: Let $A,B$ be rv's $$P(A-B\leq ...
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54 views

Difficulty finding Expectation of a special function

I have a special function given as: $${\rm f}\left(r\right) ={1 \over \beta\lambda}\,2^{r/\beta} \exp\left({\left[2^{r/\beta} - 1\right]K \over \lambda}\right)$$ I should find the Expectation of ...
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20 views

Calculating probabilities of events

Was going through past previous exam questions and came across this one: A manufacturer of lie detectors is testing its newest design. It asks 300 people to lie deliberately and another 500 people ...
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47 views

For a poker hand, five cards are chosen from an ordinary deck of playing cards. How to find the probability to get the following hands

How would you find the of probabilities: (a) a hand with 1 heart, 2 diamonds, 2 clubs (b) a hand with no face cards (c) a hand with at least 3 queens
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38 views

Probability that exists at least an edge in the configuration model

In this period, I am studying some topics on random networks to understand the modularity optimization used in community detection. In particular, I am trying to understand a model called ...
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57 views

Dice Probability4

Find the conditional probability of the event"The red one is 3, given that the sum is 6" when two fair dice (one green one red) are rolled. I got 1/6 but the answer key gives 1/5. How should I attack ...
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36 views

Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind: Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is ...
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56 views

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 52 cards?

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 52 cards? I have no clue... 6 slots, 4! ways of choosing a king and 4P2 ways of choosing a queen. I'm ...
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38 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
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1answer
32 views

How to prove this probability theory expectation problem?

Let $X$ only own values from the set $\left\{ {-n, -n+1, \ldots, -1}\right\}$ and the values may not be equally possible. Prove that then: $$E(X)=-\sum\limits_{i=1}^n P(X \leq -i)$$ Any tips are ...
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38 views

Motivation behind the proeprties of sigma algebra

What is the motivation behind the class $B$ of all measurable sets to satisfy the following properties : 1) $A_1, A_2 \in B$ implies $A_1 \cup A_2 \in B$ 2) $\{A_n\} \in B $ and $\{A_n\}$ is ...
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24 views

Convergence in probability of sample variance

$X_n$ s are a sequence off iid random variables with E($X_n$) = $\mu$, V($X_n$)= $\sigma$$^2$ and $\bar X = \sum$ $\frac{X_i}{n}$. Then show that $\frac1n$ $\sum (X_i - \bar X )^2\to\sigma^2$ in ...
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52 views

How should I define E(X|Y) if I want to show that E[E[X|Y,Z]|Z]=E[X|Z]?

I'm taking an introductory probability class and recently had a homework problem asking whether ${\bf E}[{ \bf E}[X|Y,Z]|Z]={\bf E}[X|Z]$ is true (it is). I'm trying to understand the proof for this ...
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31 views

Probability of 2 identical events

My professor said that probability of 2 identical events in a very short amount of time (dt converges to 0) is 0. However, I did not agree with him about this. Is there a proof for that assertion? ...
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38 views

Product of $n$ i.i.d. random variables

Let the variable $Z$ equal $Z = XY$ where $X$ and $X$ are two i.i.d. continuous random variables which distributions are given by $f_X()$ and $f_Y$. The distribution of $Z$ is given by: $$f_Z(z) = ...
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31 views

Problem with proving probability formula. [closed]

How to prove that $$\sum_{i=1}^nX_iP(X_i) = -\sum_{i=1}^nP(X\leq -i)$$ if X belongs to $\{-n,-n+1,....,-1\}$
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54 views

applying iterated expectation when conditioning on multiple random variables

The law of iterated expectations tells us that ${\bf E}\big [{\bf E}[X\, |\, Y]\big ]={\bf E}[X]$. Suppose that we want apply this law in a conditional universe, given another random variable $Z$, in ...
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33 views

$\mathrm{Pr}(X_1 + X_2 + \cdots + X_n \le n) .$ [duplicate]

Let $X_1,X_2,\ldots,X_n$ be a sequence of mutually independent random variables. For each $i$ with $1 \leq i \leq n$, we are given that the variable $X_i$ is equal to either $0$ or $n+1$, ...
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143 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
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162 views

Define the random variable $X$ to be the total number of distinct brands that Simon receives [duplicate]

MCBC still sells $n$ different brands of IPA. As in the previous question, when you place an order, MCBC sends you one bottle of IPA, chosen uniformly at random from the $n$ different brands, ...
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1answer
29 views

What is the distribution of the dot product of a Dirichlet vector with a fixed vector?

I am trying to get the distribution of a weighted sum when the weights are uncertain: $S = \sum\limits_{i=1}^N w_iC_i = \mathbf{w}\cdot \mathbf{C}$ where vector $\mathbf{w}$ is random with components ...
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43 views

chose uniformly at random from the n different brands, independently of previous orders [duplicate]

Michiel's Craft Beer Company (MCBC) sells $n$ different brands of India Pale Ale (IPA). When you place an order, MCBC sends you one bottle of IPA, chosen uniformly at random from the $n$ different ...
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3answers
40 views

Probability of Head in coin flip when coin is flipped two times

Probability of getting a head in coin flip is 1/2. If the coin is flipped two times what is the probability of getting a head in either of those attempts? I think both the coin flips are mutually ...
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35 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
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57 views

Product of two random variables that has exponential distribution

I am trying to define the probability distribution of $Z$ such as $Z = X_1\cdot X_2$ where $X_1$ and $X_2$ are two independent and identically exponentially distributed variables. $$P(X_1=x) = ...
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1answer
40 views

If $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$ then $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s

Let $X_t$ be an Ito's process where $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$. Prove $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s Here my solution for ...
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28 views

If $u(z),$ where $Z_t=W^1_t-iW^2_t$, is a complex anlyt. fx, show $du(Z_t)=u'(Z_t)dZ_t$

If $u(z)$ is a complex analytical function, where $Z_t=W^1_t-iW^2_t$ is a complex Wiener process, show $du(Z_t)=u'(Z_t)dZ_t$.
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1answer
30 views

Quadratic Variation for $X_t= \int \sigma_s dW_s$ where $\sigma_s \in S$

Let $\sigma_s \in S$. Setting $X_t=\int^t_0 \sigma_s dW_s$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ such that $d_n=\max_i |t^n_{i+1}-t^n_i| \rightarrow 0$ as $n \rightarrow ...
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1answer
20 views

Condition for independence of two scalar real valued random variables

I'm trying to show that given two real-valued scalar random variables $X,Y$ if $$\mathbb{P}\left(X\leq x,\, Y\leq y\right)\cdot\mathbb{P}\left(X\geq x,Y\geq y\right)=\mathbb{P}\left(X\geq x,Y\leq ...
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62 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
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1answer
48 views

Alice plays a game with $ \frac{1}{3} $ odds. Probability Question

Alice plays repeatedly a game that has three results: win, lose, or tie. Each time she plays, she wins with probability $\frac{1}{3}$ and loses with probability $\frac{1}{3}$ (and therefore she ties ...
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1answer
94 views

central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is ...
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30 views

Regarding jointly multivariate normal X1,X2…X5

So I have a question from statistical inference that I need some help with: $X_1,X_2,...,X_5$ are jointly multivariate normal with means = $\mu_i$, variances = $\sigma^2_i$, correlation = $\rho$ ...
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1answer
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Exchangeability with random effects?

Consider a $N\times N$ random matrix $$ \epsilon:= \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \dots &\epsilon_{1N} \\ \epsilon_{21} & \epsilon_{22} & \dots & ...
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A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
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Quadratic Variation for $X_t= \int b_s ds$ where $b_s$ is an $F_t$ adapted process.

Let $b_S$ be an $F_t$ adapted process, Borel measurable in $t$ st $\int |b_s|^2ds < \infty$ (a.s). Setting $X_t=\int^t_0 b_sds$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ ...
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44 views

A basic problem on weak convergence

Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that ...
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14 views

find number of elements combinations covered by a given set of element groups

Here is the problem I have faced recently that I cannot deal with yet and I need some help: Given is the list of elements (numbered): e.g. [1,2,3,4,6,7,8] the count and size of groups, which can ...
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63 views

How to numerically evaluate the CDF of this random variable?

I have a discrete random variable $X=0,1,2,\ldots$ with the following probability mass function: $$P(X=x)=\sum_{t=0}^x\binom{n}{t}p^t(1-p)^{n-t}\binom{m-t}{x-t}q^{x-t}(1-q)^{m-x}\tag{1}$$ where ...
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18 views

I. i. d distributions as best car offers

I was reading my probability text book and got stuck on what seemed to be a easy question: Let $X_0,X_1,\ldots,$ be i.i.d and imagine they are the offers you get for a car you are going to sell. Let ...
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16 views

A basic question on weak convergence

I have to give an example of a sequence of distributions $\mu_{n} => \mu$ (weakly sense) but $\mu_n(A) -> \mu(A)$ fails for some $A$ and all the distributions $\mu_n$ and $\mu$ are coming from ...
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35 views

probability? Marbles question

A bag contains two blue marbles and two red ones; two marbles are drawn at random. (a) What is wrong with the following argument? Because there are four possibilities – (red, red), (blue, blue), ...
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33 views

Dependence between a joint probability distribution

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...
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27 views

hat problem and probability

There are 7 prisoners in the room. In the entrance all of them get hat in one of random 2 colors: white or black. They are sitting in the circle and the light on. All of them see hat color of the rest ...
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1answer
65 views

Can a countably generated $\sigma$-algebra be “approximated” by a $\sigma$-algebra generated by a countable partition?

My question is a bit vague, hopefully someone can still clarify. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and assume that $\mathcal F$ is countably generated. My question is, does ...