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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3
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1answer
177 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
4
votes
1answer
152 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
2
votes
0answers
40 views

Calculating $\mathbf{P}[X < Y]$ for $X, Y$ exponentially distributed?

This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course« Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain ...
0
votes
1answer
25 views

An equality from the well-known analysis of variance formula

Suppose that we have a parametric model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ dominated by some measure $\mu$. That is, each $\theta$ is associated with a density $l(y;\theta)$. Let $S(Y)$ ...
1
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0answers
53 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}-\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...
3
votes
0answers
88 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
1
vote
1answer
73 views

Unbounded stopping time

Suppose we have a sequence of i.i.d. random variables $(X_n)_{n \in \mathbb{N}}$ with $\mathbf{P}(X_n = -1) = \frac{1}{2}, \mathbf{P}(X_n = 0) = \frac{1}{3}, \mathbf{P}(X_n = 1) = \frac{1}{6}$. Denote ...
1
vote
2answers
45 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
4
votes
1answer
138 views

Exercise in Probability/Measure Theory

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let also $A_{n,j}\in\mathcal{F},n\in\mathbb{N}_0,j\in\{1,2,3,...,2^n\}$, be such that for all ...
0
votes
0answers
49 views

Create a Martingale out of a Markov Chain.

Consider a homogeneous finite state Markov chain $\{X_n\}$ with transition matrix $P$ and state set $S$ consisting of real numbers. How to choose the elements of $S$ so that $\{X_n\}$ be a ...
1
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0answers
29 views

About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
1
vote
0answers
25 views

Convergence in probability of the Fisher information

Given a family $\{\mathbb{P}_\theta\}_{\theta\in\Theta}$ on $\mathcal{B}(\mathbb{R})$, where $\Theta\subset\mathbb{R}$ and each member of this family is absolutely continuous w.r.t. $\lambda^1$, and ...
0
votes
1answer
28 views

Does law of total probability apply here?

Let $X$ and $Y$ by positive independent random variables. Let $f(x,y)=\frac{ax}{y^2+ay}-\frac{ab}{y}$ where $a>0$ and $b>0$ are constants. I am wondering if the following is true: ...
2
votes
1answer
25 views

Condition in a theorem in Probability theory.

I passed by a simple theorem in Probability theory , yet it really bugs me that I think that 1 condition in the hypothesis is not necessary. After checking the proof for many times, I still can't ...
0
votes
1answer
200 views

Expected value of a complicated function of more than one random variable.

Assume we have random variables with Probability Density Functions (pdf) as follows $$\omega_i \sim f_{1},\,\,\,\,\ i \in[1:n]$$ $$ \gamma= \{\gamma_1,\cdots,\gamma_n\} \sim f_2: \text{joint pdf of ...
0
votes
0answers
25 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
0
votes
1answer
73 views

PDF of distance from the center of a random point in the unit disk

I found in a certain website (also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: $P(d)=2d$, but no one is giving the way to derive ...
3
votes
1answer
94 views

Exercise on Martingales

I have been struggling with the following exercise and I was wondering whether my solution is correct or not. I am pretty sure about the second part of the question (the martingale part) but not so ...
1
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0answers
35 views

Question about Lebesgue Dominated Convergence Theorem involving a Markov Time / Stopping Time

I am trying to understand the proof of the following lemma: Let $W$ be an arbitrary random variable satisfying $\mathbb{E}[|W|] < \infty$, and let $T$ be a Markov time (or stopping time) for which ...
0
votes
1answer
35 views

Dependence of RVs exponentially distributed

Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in ...
1
vote
1answer
27 views

Does $P(X\leq a) = P(X^2\leq a^2)$ if $X$ is a positive random variable and $a>0$?

The answer looks positive to me, since $$P(\omega:X(\omega) \leq a) = P(\omega:X(\omega)^2\leq a^2)$$ Am I right?
1
vote
1answer
40 views

Characteristic function, how to integrate?

Find the characteristic function $\phi_X(t)$ of an absolute continious r.v. $X$ with density: $$f_X(x) = \frac{a}{2}e^{-a|x|} \qquad (a>0; x\in \mathbb{R})$$ Notation I have some ...
1
vote
0answers
31 views

given distribution function find the density

I am confused about one trivial thing. We have independent variables $X,Y,\beta$ where $X$ has a distribution function $F(x)$, $Y$ has $G(x)$ and $\beta$ is $Bin(1,p)$ distributed. Also $X,Y$ have ...
2
votes
0answers
52 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
3
votes
1answer
61 views

Limit of measures is a measure

I know the following theorem (see exercise 1.3.3 from Achim Klenke: »Probability Theory — A Comprehensive Course«): Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of finite measures on the ...
0
votes
0answers
37 views

What is the interval for an exponential random variable?

Suppose that I have generated a random number $x$ using an exponential distribution with rate parameter $\lambda$. How can I find an interval $[a,b]$ such that $x$ is in this interval with probability ...
3
votes
5answers
150 views

Conditions for uniqueness of the median

A median of a random variable is defined as any $m \in \mathbb{R}$ such that $P(X \le m) \ge 1/2$ and $P(X \ge m) \ge 1/2$. Alternatively, in terms of the CDF $F$ of $X$ defined by $F(x) := P(X ...
0
votes
1answer
30 views

Conditional probability -conditioning on a random variable

Let $ (\Omega, \mathcal{F}, \mathbb{P}) $ be a probability space, $A \in \mathcal{F}$ and $X$ a random variable. What does it mean $$ \mathbb{P} (A | X) $$ when $X$ is not discrete? Thank you!
5
votes
1answer
45 views

Does this sequence converge almost surely or not?

I have a sequence of independent random variables $X_1, X_2,...$ such that $P(X_n = 1) = \frac{1}{n}$ and $P(X_n = 0) = 1 - \frac{1}{n}$. Using the second Borel-Cantelli Lemma, we have $\sum P(X_n ...
2
votes
1answer
23 views

Clarification - DeGroot Proof on Transitivity Property of Subjective Probability

In developing axiomatic foundation for subjective probability DeGroot (Optimal Statistical Decision, 2004, p71) gives two axioms/assumptions: SP1: For any two events A and B, exactly one of the ...
3
votes
3answers
77 views

Probability that a size $d$ sample will contain all $k$ colours present

I tried looking for this question, but couldn't find it exactly... apologies if this is a repeat! Imagine an urn with $m$ balls. Each ball has a different colour and there are $k$ colours (obviously, ...
2
votes
1answer
34 views

Expectation of maximum of Binomial RVs

Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find $$E(X_{(n)})$$ that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that ...
0
votes
2answers
29 views

Finding probabilities - Elementary probability.

Suppose $\mathbb P(A) = 0.3$, $\mathbb P(B) = 0.5$, and $\mathbb P(B |A) = 0.6$. a. Find $\mathbb P(A \text{ and } B)$. b. Find $\mathbb P(A \text{ or } B)$. c. Find $\mathbb P(A|B)$. ANSWER: a. ...
1
vote
1answer
83 views

Conditional independence of sigma-algebras

If ${\mathcal{H}_1}$ and ${\mathcal{H}_2}$ are conditionally independent given $\mathcal{G} \subseteq {\mathcal{H}_2}$, are they conditionally independent given $\mathcal{F}$ such that $\mathcal{G} ...
0
votes
1answer
120 views

What does (0+) mean?

I'm currently learning from a script (which is written in German and not publicly available, sorry) for introduction to stochastics, where the topic is the Laplace transformed function for random ...
0
votes
1answer
47 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...
1
vote
0answers
47 views

A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
1
vote
1answer
14 views

Expectation of distance from centre of a circular scattering

A random point $(X,Y)$ has a normal distribution on a plane with circular scattering with $E[X]=E[Y]=0$ and var$[X]$=var$[Y]$=$\sigma^2$. The distance of the point $(X,Y)$ from the centre of ...
2
votes
1answer
84 views

Probability Of 2 consecutive dice throw for 4 players

Consider a fair dice. 4 players are throwing the dice one after another. If some one gets 6 he will get extra chance to throw it again. So he will throw it twice and then the next player will get the ...
3
votes
1answer
64 views

Natural Filtration and Sigma-Field Generated by path function

Suppose we have a continuous real-valued stochastic process $X=(X_t;t\geq 0)$ defined on a probability space $(\Omega,F,P)$. Usually one defined the filtration to be $F_t=\sigma(X_s;s\leq t)$. But on ...
1
vote
1answer
38 views

How do you represent a random choice of random variables mathematically? What is its mean, variance, etc.?

Suppose that I have six random variables $X_1, X_2,\ldots,X_6$ (say, e.g., six coins with different biases). We should be able to get a new random variable $Y$ by rolling a die to get a number $n\in ...
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votes
1answer
51 views

Confusion about calculating probability of at least one event occurring

The probability that Tom will win the Booker prize is 0.5, and the probability that John will win the Booker prize is 0.4. There is only one Booker prize to win. What is the probability that at least ...
2
votes
2answers
58 views

Estimating $\mathbb P\{\max_{1\le j\le n}\lvert S_j\rvert\le t\}$, so called Charles Stein's theorem?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.5, Ex6) Suppose $\{X_n\}_{n>0}$ is a sequence of i.i.d. random variables. $S_n:=X_1+\dotsb+X_n$. For each $t>0$, define ...
1
vote
1answer
38 views

Normal approximation of binomial distribution

Problem: On average, every 50th shell has a pearl. What is the minimum amount of shells you have to open to get at least one pearl with probability greater or equal to 0.95. Calculate using the ...
0
votes
0answers
18 views

Partial derivative of a random vector

If $x$ indicates a $1\times n$ random vector of any distribution, then is the partial derivative of $x$ w.r.t $x$ equal to the derivative of the individual elements in the matrix, or are they just the ...
3
votes
1answer
29 views

Martingales and different definitions

Are there any differences between the following definitions of Martingales and if so what are they? Let $(X_{i})_{i=1}^{n}$ and $(Z_{i})_{i=1}^{n}$ be sequences of random variables then $(X_{i})$ ...
1
vote
1answer
94 views

Find the characteristic function of combination of random variables

I have the same problem as here: Find characteristic function of random variable. Could you explain last equality? Can I get it without using law of total expectation? Update I have some idea, $E $$ ...
0
votes
0answers
38 views

Cramer-Rao-Bound for squared parameter

I have a random sample $X_{i}, \ldots, X_{n} \sim \mathcal{N}(\theta,1)$. I would like to find the Cramer-Rao lower bound for the variance of unbiased estimators of $\theta^{2}$. Here's what I have: ...
2
votes
0answers
53 views

Expected magnitude of a vector of $n$ i.i.d. random variables as $n\to\infty$

Suppose that $X_i$ are i.i.d. real valued random variables with probability distribution $f(x)$ for $i=1,2,3,\ldots$. Let $Y_n=\left(\sum_{i=1}^nX_i^2\right)^{1/2}$. Assuming that ...
2
votes
1answer
147 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...