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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About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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57 views

Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ ...
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12 views

Does $(X⊥Y | z^0)$ for binary-random variable Z implies $(X⊥Y | Z)$?

I have been attempting to reason about this question. Does $(X⊥Y | z^0)$ for binary-random variable Z implies $(X⊥Y | Z)$ ? My intuitive thought is that this is not possible but I would really ...
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36 views

Event-level independence does not imply random-variable independence for non-binary random variables

A single event-level independence $(x^0⊥y^0)$ does not imply the random variable level independence $X⊥Y$ for non-binary random variables. By non-binary I mean, random variables has more than two ...
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3answers
34 views

Independence of max and min of a set of random variables.

Suppose $X_1,\ldots,X_n$ are independent and identically distributed random variables with cdf $F_X(x)$. Define $U$ and $L$ as $U=\max\{ X_1, \ldots ,X_n\}$ and $L = \min\{X_1,\ldots,X_n\}$. Are $U$ ...
1
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1answer
42 views

Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
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1answer
13 views

Probability using Chebyshev equation

A dice is rolled until the result is 6 and this is repeated n times. Sn represents the total amount of times the dice is being rolled (in order to get 6 total of n times). Find the probability for ...
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1answer
24 views

Show if $X,Y$ are independent RV's (discrete or continuous) then $k+X,Y$ and $kX,Y$ are indepedent for $k \in \mathbb R$.

Show if $X,Y$ are independent RV's (discrete or continuous) then $k+X,Y$ and $kX,Y$ are indepedent for $k \in \mathbb R$. I've been thinking how to prove the above statement. Intuitivelly it's ...
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2answers
19 views

probability related to uncountable set

Let $X$ be the collection of closed interval of the form $[a,1],$ where $a \in [0,1]$ and we fix a real number $t \in [0,1]$. Suppose an element $c$ is randomly drawn from $X$, what is the probability ...
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1answer
32 views

Let $(X,Y)$ be an absolute continuous R.V. Find $XY \mid Y = y$ and show $XY$ and $Y$ are independent. Also $XY \sim e(1)$.

Let $(X,Y)$ be an absolute continuous R.V with density $f_{X,Y}(x,y) = ye^{-y(x+1)}, \ x,y >0$. I've shown that $Y \sim e(1)$ and $X \mid Y = y$ density $x \mapsto ye^{-yx}$. However I must find ...
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1answer
22 views

Product of two random variables together with conditional density

Let $X_1$ and $X_2$ be two real valued random variables such that we have the conditional density of $X_1$ given $X_2$, i.e. $$\mathbb P(X_1\in M\mid X_2) = \int_M \phi(x_1\mid X_2)dx_1$$Also, let $h$ ...
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1answer
47 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
2
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1answer
29 views

Show that $\dfrac{\max(X_1,\cdots,X_n)}{n}\xrightarrow{a.s} \,0$.

Let $\{X_n\}$ be a sequence of identically distributed random variables with $\text{E}(|X_1|)< \infty$. Show that $\dfrac{\max(X_1,\cdots,X_n)}{n}\xrightarrow{a.s} 0$ I need to show ...
2
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3answers
50 views

Why can we use entropy to measure the quality of a language model?

I am reading the < Foundations of Statistical Natural Language Processing >. It has the following statement about the relationship between information entropy and language model: ...The ...
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9 views

What does it mean for parametric model to be smooth?

How "smooth" parametric model is defined? Is it smooth w.r.t. parameter? If the parametric model is ${\cal Q}=\left\{q_{\theta}|\theta\in\Theta\right\}$ with densities $q_{\theta}$, is smoothness ...
3
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25 views

Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?

Volume forms on smooth manifolds have a nice interpretation as measures, but what takes the place of the Borel $\sigma$-algebra? In particular, if we let $\mathcal{M}$ be a smooth manifold and ...
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3answers
75 views

probability question involving numbers 100 to 999 inclusive

A whole number between 100 and 999 inclusive is chosen at random. Find the probability that it is exactly divisible by 3. If it is exactly divisible by 3, what is the probability that it is exactly ...
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1answer
19 views

Empirical characteristic function

The ecf is $\phi_n(\omega) = \frac{1}{n}\sum_{j=1}^ne^{iX_j\omega}$. I'm stuck on trying to see why the following is true $$|\phi_n(\omega)|^2 = \phi_n(\omega)\phi_n(-\omega)$$ Wouldn't this imply ...
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2answers
47 views

Is this probability inequality always true?

For $n$ random variables $X_1$, $X_2$, $\dotsc$, and $X_n$. Is it always true that: $$\mathbb{P}\left[\sum_{k=1}^{n} X_k>a\right]\geq\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}>a\right].$$ ...
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14 views

A question regarding the probability density function of a uniformly distributed random variable.

In lecture, my teacher stated that the probability density function of a uniformly density random variable is not continuous. I can't seem to figure out why that is. Can anyone explain why?
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19 views

How to calculate this CDF?

Let suppose that we have three points in the euclidean plan $\mathbb{R}^2$ which are depicted inside a circle of radius $R$ as follow: $P_1=(D,0)$ (the center of the circle), $P_2=(0,0)$, and ...
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1answer
45 views

How to get $E(X^2)$ through $E(X)$?

Here's my exercise: A fair dice is thrown $n$ times till the result is six. $S_n$ is the sum of the numbers of necessary throws. Using Chebyshev's inequality, find an estimation to the probability ...
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1answer
23 views

Probability theory: calculating the probability of a joint event

I want to determine the probability of a 'joint' event. The events are $A,B$ which are independent. $A$ is Binomial distributed with pmf $p_A(A)$ and $B$ is given by its cdf $P_B(B)$ and is also ...
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55 views

Probability Theory - Fair dice

Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. Im unsure ofwhat type of question this is, I have tried ...
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34 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
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1answer
33 views

Is probability a concept or a number?

I recently had a discussion with a friend about the possibility that the earth is supported on the back of a giant tortoise. My position was that it definitely isn't so. I pointed out that like ...
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1answer
15 views

Number of storms in a rainy season

This is a follow-up to my previous question. Now instead of finding a probability I would like to now find the expectation too. I will restate the question and my solution below. I would appreciate if ...
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1answer
6 views

Probability given pairs

I have a list of numbers (1 - p) except I have every number twice. Given a random subset of these numbers (say length q). What is the probability that I have a duplicate number? Thanks
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35 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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27 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
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1answer
66 views

Levy's Martingale Using Radon Nikodym

Let P and Q be two probability measures on the same space $(\Omega,\mathcal{F},\mathcal{P})$ and let $\mathcal{F_n}$ be filtration. Assume that $Q \ll P$. Let $X_n$ denote the Radon-Nikodym derivative ...
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32 views

Helly-Bray theorem for weak convergence

Let $\{\mu_n\}_{n \geq 1}, \mu$ be probability measures on $(\Bbb R, B(\Bbb R))$. Then I need to prove that $\mu_n => \mu$ implies $\int fd\mu_n -> \int fd\mu$ where $f$ any bounded and ...
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45 views

Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
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Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
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26 views

Interchanging limits in stochastic order notation definition

A sequence of real-valued random variables $(X_n)$ is said to be $O_P(b_n)$ where $(b_n)$ is a sequence of positive numbers if $$ \lim_{T\to\infty} \limsup_{n\to\infty}P\{|X_n|>T b_n\}=0$$ Is it ...
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23 views

Help with random variable to found probabilty (PDF)

Stuck in this example to found (PDF) in many conditions
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1answer
26 views

I stuck in binomial probability (PMF) with parameter n & p

I stuck in this example, but I have many trying
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1answer
31 views

The probability of the first roll having the highest number in n consecutive rolls of one 6-sided die

What is the probability that if we roll one six-sided die for n times - whatever the number we see in the first roll, no other following roll has a number higher than that (that is all the numbers we ...
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1answer
56 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
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1answer
22 views

Conditional Expectation on similar sigma algebras

I'm trying to prove the following, (or to find a counterexample): Let on a probability space. $Y$ be a Bernoulli variable, $X\in L^1$ be another random variable, let $\mathcal{G}$ be some ...
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15 views

Expectation of O_p(1) process

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$. Is there any condition(s) to guarantee that ...
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3answers
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Regarding the validity of probability theory [closed]

Imagine I have a regular balanced dice and i roll it once. It is assumed that the probability of any number (1-6) is 1/6. However, isn't this just an illusion we are feeding ourselves for our lack of ...
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1answer
34 views

Fixed-time Jumps of a Lévy process

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I ...
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27 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
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1answer
43 views

Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a ...
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1answer
42 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
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23 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
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30 views

Question about computing the sample mean and variance values from a sample coming from a Weibull Distribution …

Let's suppose that I have a random sample x from a Weibul distribution with shape parameter k=1 and scale parameter λ=2... How am I supposed to compute the mean value of the sample ? Also what can I ...
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24 views

Probability using sup

I have to prove that $$P(A\cup B)=P(A)+ P(B)$$ if $A\cap B=\emptyset$. Using the definition of sup $$P(A)=\sup \left\{\sum_{x_n\in J}p_n:J\subset A, J\ finite\right\}$$ But i really don't know how to ...