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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83 views

Probability of $P(X>Y)=p_1, P(Y>Z) = p_2, P(Z>X) =p_3$, find minimum of $p_1 +p_2 +p_3$

$X,Y$ and $Z$ are $3$ random variables equalling numbers and a probability is defined such that: $P(X>Y)=p_1,\, P(Y>Z) = p_2, \,P(Z>X) =p_3$ a) thus find maximum of $p_1 +p_2 +p_3$ b) ...
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155 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
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1answer
163 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
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2answers
136 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
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57 views

An application of LDCT

Consider the sequence of functions: $$F_n (t)=\int_{-\infty}^t \underbrace{ \frac{\Gamma \left[ \left( n+1 \right)/2 \right] }{\sqrt{\pi n} \Gamma \left(n/2 \right)} \frac{1}{\left(1+y^2 /n ...
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1answer
48 views

How $\Pi_{i=0}^n\xi_i$ converges a.s. to $0$ provided $\xi_n>0 $, iid and $E(\xi_n)=1$

suppose $\{\xi_n,n \ge 0\}$ are iid and positive random variables. $E(\xi_0)=1$. show $\Pi_{i=0}^n\xi_i$ is a positive martingale converging to $0$ provided $P[\xi_0=1]\not=1$ It's easy to prove ...
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365 views

A simple probability of the elevator's stop

There is a 10 floors building, 10 people get in the elevator in the ground floor and get off at each floor independently. What is the probability that the elevator stops at floor 5? My answer is ...
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1answer
112 views

Characteristic function of an r.v. with finite variance and zero mean.

Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
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579 views

Let (X,Y) have a Dirichlet Distribution with paramters $(\alpha_1, \alpha_2, \alpha_3)$ Establish that X~Beta$(\alpha_1, \alpha_2 + \alpha_3)$

If the joint pdf of (X,Y) is $f(x,y)=\frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} x^{\alpha_1 - 1} y^{\alpha_2 - 1} (1-x-y)^{\alpha_3 -1}$ ...
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146 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
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146 views

measurability question with regard to a stochastic process

Here are two related exercise from Karatzas and Shreve Let $X$ be a process, every sample path of which is right continuous with left limits. Let $A$ be the event that $X$ is continuous on $[0,t_0)$. ...
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896 views

Tossing a biased coin until heads comes up, Find $P(X=n)$ and $P(X\leq n)$

A biased coin that comes up heads with probability p, is tossed until the first head appears. Let random variable X be the number of tosses. For a fixed $n \in N$ find $P(X=n)$ and $P(X\leq n)$. My ...
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45 views

Got Stuck with these probability problems

I tried my best to solve 'em , but after waiting a few sheets of paper , I got nothing on me . A litle help from you guys might do the trick , Thanks ! ...
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166 views

Probabilistic puzzle

There are $n+1$ boxes and every box contains $n$ balls. For every $k\in\left\{ 0,1,\ldots,n\right\} $ there is exactly $1$ box containing $k$ white balls and $n-k$ black balls. A box is picked out and ...
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327 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
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1answer
67 views

expectation of (largest) number of consecutives

Go out from independent events $E_{1},E_{2},\ldots$ that can succeed or fail. This with $p=P\left\{ E_{n}\text{ succeeds}\right\} $ for each $n$. For $n=1,2,\ldots$ define $X_{n}$ as the largest ...
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1answer
210 views

Mean and variance of geometric function using binomial distribution

Can anyone help solving this question please? I tried but not sure of the steps to reach the conclusion.
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654 views

Borel sigma field

Is the $ \sigma $ - field generated by $[a,b]$, $a,b \in \mathbb Q $ and $ \sigma $ - field generated by $(a,b)$, $a,b \in\mathbb Q^c $ identical? Are they also the same as Borel $\sigma $ - field. ...
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1answer
50 views

Proof this form

I am trying to proof this form: Let $g(.)$ be a function, for $y_n$ is a a nonnegative random variable, $\varepsilon>0$, $g(x)>0$ is increasing function for $x>0$, and $E[g(x)]>0$, then ...
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1answer
102 views

finding a generating function of a gambler question

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability p and loses one dollar with probability 1 - p. Let fn be the probability that he or she fi ...
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3answers
61 views

the number of components working at a particular time

Suppose a system has $10$ components and that a particular time the $j$'th component is working with probability $1/j$ for $j=1,2,\dots,10$. How many components do you expect to be working at that ...
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273 views

Expected number of light bulbs on

There are 50 light bulbs in a room each with its own switch. Initially all light bulbs are off. Dick follows the following procedure 50 times: He randomly selects a light bulb and flips the switch, ...
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94 views

On continuity of measure

Let $m$ be a probability measure on $\mathbb{R}^n$. Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$. Say under what conditions the following inequality holds. $$ m\left(\left\{ x \in ...
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1answer
2k views

How to compute conditonal probability on multiple conditions on the right side?

Is there a formula for a conditional probability of many "conditions" I.e: $P(X|Y,Z,\ldots)$ That is how to compute $P(X)$ given $X,Y,\ldots$ Or given separately $P(X|Y)$ and $P(X|Z)$ is there a ...
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60 views

Help with an asymptotic proof?

I am having difficulty with a proof of an asymptotic result in maximum likelihood estimation theory. I don't believe any statistical theory however is needed to solve this problem, rather I think ...
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1answer
95 views

Support vs range of a random variable

Is there any difference between the two? I have not met any formal definition of the support of a random variable. I know that for the function $f$ the support is a closure of the set ...
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1answer
123 views

Continuity of a probability integral with supremum

Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1] $ be a probability measure without point masses and let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be continuous. Let $\epsilon ...
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1answer
90 views

Understanding Conditional Expectation

I just want to make sure I'm understanding conditional expectation correctly: Let $X_1,X_2,X_3$ denote three independent coin flips with probability of heads $\frac{1}{4}$ and probability of tails ...
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95 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
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72 views

How does this violate probability theory?

Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$) $p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$ $p(Y = -1) = .5$, $p(Y = 3) = .5$ Question: Despite not being handed any information ...
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82 views

$\mathcal{A}\perp_\mathcal{G}\mathcal{B}\wedge\mathcal{H}\subseteq\mathcal{G}\implies\mathcal{A}\perp_\mathcal{H}\mathcal{B}$?

If $\mathcal{A}\perp_\mathcal{G}\mathcal{B}$ and $\mathcal{H}\subseteq\mathcal{G}$, is it the case that $\mathcal{A}\perp_\mathcal{H}\mathcal{B}$? Here $\mathcal{A}$, $\mathcal{B}$, $\mathcal{G}$ and ...
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282 views

Tail events and exchangeable events

In this problem I have $X_1, X_2, \cdots$ independent identically distributed RVs taking values $\pm1$ with the equal probability of $1/2$ and my trajectory is defined by $S_n=\sum_i^n X_i$ (so pretty ...
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1answer
157 views

Conditional Probability Proof

Suppose that X and Y are independent discrete random variables. Let h(x,y) be a bounded two-variable function. Show that: E [h(X,Y)|X = x] = E [h(x,Y )] Explain why this is usually not true if X and ...
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1answer
118 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
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521 views

Definition of atomic $\sigma$-field.

Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
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1answer
82 views

How does Lindeberg CLT imply classicial CLT?

For a sequence of zero-mean $\sigma^2$-variance IID random variables $X_1, \dots,$ the Linderberg condition is applied this way: $\forall \epsilon >0$, as $n$ goes to $\infty$, $$\sum_{i=1}^n ...
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353 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
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1answer
43 views

Simple probability system

A system consists of four components ${1,2,3,4}$. There are two routes ${1,2,4}$ and ${1,3,4}$ to complete the system. If all four components have equal reliability of $0.9$, what is the reliability ...
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154 views

Is the product of two mixing random variables also mixing?

Question: Assume $X_t$ and $Y_t$ are random variables from the same probability space adapted to the filtration $\mathcal{F}_{-\infty}, ..., \mathcal{F}_t, ..., \mathcal{F}_{\infty}$. If $X_t$ and ...
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84 views

Relation between support of image-measure and closure of the image

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a measurable map. For a probability measure $\mathbb P$ denote by $\mu_\mathbb P$ the image measure of $\mathbb P$ ...
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2k views

Probability that numbers 1…6 show up at least once when rolling 8 dice

Probability that numbers 1...6 show up at least once when rolling 8 dice How can this be solved using the inclusion-exclusion principle.
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228 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
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147 views

Probability, Poisson Process, My solution is correct?

Jobs are submitted for a computer according to a Poisson process with a rate λ (jobs / hour). Determine the probability of at least two jobs are submitted within the 'a' first few minutes. I tried ...
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1answer
84 views

How is $P(|A) $ defined from $P(|\mathcal{B})$?

Let $ (\Omega, \mathcal A, \operatorname{P})$ be a probability space, with a real random variable $X$ and a sub-σ-algebra $ \mathcal B \subseteq \mathcal A$ In probability theory with measure ...
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1answer
124 views

Lévy-Khintchine formula for Cauchy distribution

The Lévy-Khintchine formula for the log of the characteristic function of an infinitely divisible random variable is $$ \Psi(s)=ias + \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} + ...
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74 views

Geometric distribution, tossing a die [duplicate]

Possible Duplicate: geometric distribution throwing a die Yesterday I posted a question which was answered but I disagree with the answer so I'd like to ask again so we can discuss it ...
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2answers
209 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
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263 views

Sequence of martingales

Let $(X_n)$ be a sequence of independent, identically distributed random variables with finite moment-generating function $M(t) = \mathbb{E}\left[\exp(tX_1\right)] < \infty$ for $t \in \mathbb{R}$. ...
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1answer
221 views

Poisson distribution and probability of random variables

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
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156 views

Why is the following example of a Markov process not strong Markov

$X(t) := 0 \;\; (t \leq \tau),\;\; t - \tau\;\;(t \geq \tau)$ with $\tau$ exponentially distributed. Then X has the Markov property but not Strong Markov Property. But why ???? Can someone kindly ...