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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
41 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
1
vote
2answers
79 views

Probability with powers of $2$

There are $2^H$ tickets. Let $k<H$ (and so $2^k$ is a submultiple of $2^H$). Let $D$ be a random integer number taken uniformly from $1,2,..k$. Then $2^D$ is in turn a submultiple of $2^k$ and ...
1
vote
1answer
104 views

Change the order of conditional expectation of integration

I encountered this problem when learning SDE: $g(t,\omega)$ is a adapted process then $$\mathbb E\left(\int_a^b |g(t)|^2 \, dt \mid \mathcal F_a\right)=\int_a^b\mathbb E\left(|g(t)|^2\mid \mathcal ...
1
vote
0answers
41 views

Independence of increments of some processes

I am stuck on this question: Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$ Any ideas how ...
1
vote
1answer
53 views

Joint density invariant under orthogonal transformations [duplicate]

I have a problem and I am totally stuck! I have to show that when the distribution of two random variables $X$ & $Y$ given by $g(x,y)=f(x)f(y)$ is invariant to orthogonal transformations, then it ...
1
vote
1answer
77 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
1
vote
1answer
40 views

How to give a good upper bound on tail probability for $P\{|\frac{R_n}{\sqrt{n}}-1| \ge \varepsilon\}$?

Suppose $X_1,X_2,\ldots$ is a sequence of i.i.d. standard normal random variables. $R_n=\sqrt{X_1^2+\ldots+X_n^2)}$. How could I prove $P\{|\frac{R_n}{\sqrt{n}}-1| \ge ...
1
vote
0answers
37 views

Is set of product distributions compact under second moment constrains?

Please do not treat this question as duplicate of Definition of the set of independent r.v. with second moment contstraint which I didn't want to edit because of many useful comments. Also, in this ...
1
vote
1answer
41 views

Set of product distribution is the cross product of its sections

Let $(X_1,X_2)$ be an independent random pair with distribution $F(X_1,X_2)$. Let \begin{align*} S&=\left\{F(X_1,X_2)\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E[X_1^2] \le 1, \ E[X_2^2] \le 1 ...
1
vote
0answers
117 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
1
vote
1answer
170 views

If $U$ is uniformly distributed on $S^{2}$, then its first component is uniformly distributed on $(-1,1)$.

Assume $\mathbf{U} = (U_{1}, U_{2}, U_{3})' \sim unif(S^{2})$. How would I show that $U_{1} \sim unif(-1,1)$? I don't know if I'm confusing myself, because I can't see this as being true for higher ...
1
vote
1answer
58 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty ...
1
vote
1answer
49 views

Independence and uncorrelatedness between two normal random vectors.

If $X$ and $Y$ are normal random vectors in $\mathbb R^n$ and in $\mathbb R^m$, and they are jointly normally distributed i.e. $(X,Y)$ is normally distributed in $\mathbb R^{n+m}$, then are the ...
1
vote
1answer
64 views

Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not generally true that $E\left(e^X\right)=e^{E(X)}$ (we can only use Jensen's Inequality here). ...
1
vote
0answers
33 views

Convergence of a distribution function

I have a problem that has me stumped. Frustratingly enough I can't really get the problem started. It goes as follows: Let $X_{1},X_{2},...$ be independent $C(0,1)$-distributed random variables. ...
1
vote
1answer
77 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
1
vote
1answer
62 views

Composition of random variable with its distribution is uniform

I'm trying to solve the following problem (Exercise 11.13) from Probability Essentials by Jacod and Protter: Let $X$ be a random variable (on $\mathbb{R}$) with distribution $F$ that is continuous. ...
1
vote
1answer
40 views

A question about random walk similar Markov Chain

This is an exercise from Probability and Measure by Billingsley: Let $(X_n)$ be a Markov chain with $\Bbb{Z}$ as its state space and with transition probabilities $(p_{ij})$ such that $$ ...
1
vote
2answers
151 views

Expectation of Truncated Random Variables

Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $0<\delta<0.5$ and $\epsilon >0$ and define ...
1
vote
1answer
50 views

Finding $E(X \mid X > Y )$ when $X, Y \sim N(0,1)$ [duplicate]

This is the problem: The random variables $X$ and $Y$ are independent and $N(0,1)$-distributed. Determine $E(X \mid X > Y )$, $E(X + Y \mid X > Y )$. I go by the definition ...
1
vote
1answer
49 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
1
vote
1answer
53 views

Literature request on Markov Chains which state transition probability matrix evolves over time

I want to know is there any literature on markov chains who's state transition probability matrix evolves over time? For instance, I have 2 states, 1 and 2. With $$P = \begin{bmatrix} p_{11} & ...
1
vote
1answer
128 views

Proof about Inclusion-exclusion formula?

The problem requires to use the indicate function to prove the Inclusion-exclusion formula. But I really don't know what to do. Anyone can help with that? Thanks!
1
vote
1answer
55 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
1
vote
0answers
53 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
1
vote
1answer
73 views

Probability Question: Bridge problem

There are $n$ islands in the ocean. Each island is linked by a single bridge between each and every unique pair of islands to ensure no island is isolated from the others. The probability of each ...
1
vote
1answer
382 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
1
vote
1answer
46 views

Random variable bounded by another random variable

How to find $\Pr(z<X<Y)$ if $X$ and $Y$ are independent exponential r.v.'s with parameters $\lambda$ and $\mu$ So $x$ is bounded by $z$ and $y$, and y must be go from $z$, (and not from ...
1
vote
0answers
62 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
1
vote
0answers
31 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
vote
1answer
70 views

Two notions of total variation norms

I found these two definitions of the total variation norm for probability measures on $(X,\mathcal{F})$: $$ \left \|\mu- \nu \right \|_{TV} = \sup_{\text{$f:X \rightarrow [-1,1]$ measurable}} \left ...
1
vote
1answer
85 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
1
vote
1answer
46 views

Characterization of conditional independence

Definition: Let $\mathcal{G},\mathcal{K},\mathcal{H}$ be $\sigma $-subalgebras of $\mathcal{F}$, where $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ is a given probability space. We say that ...
1
vote
2answers
55 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
1
vote
1answer
79 views

Product measure with a Dirac delta marginal

Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like ...
1
vote
1answer
100 views

Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
1
vote
1answer
186 views

Expectation and variance of correlated exponential brownian motions

What is the expectation and variance of correlated exponential Brownian motions for the random variable $F$, where $A$ is real constant, $\sigma$ is a real constant and $\rho$ is the correlation. $$F ...
1
vote
2answers
303 views

Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution …

Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$. GOAL: Show that ...
1
vote
0answers
120 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
1
vote
1answer
39 views

Expected value of a sum starting at a value given through a random variable

I've got a question concerning the expected value of a sum which starts at a certain value given through a random variable. More precisely: $$G(n):=P[X \geq K]$$ where $X \in Bin(n,p)$ and $K$ is ...
1
vote
0answers
156 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
1
vote
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 ...
1
vote
1answer
70 views

Random operators

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
1
vote
1answer
63 views

Some property of conditional independence

Given random variables $Y, Z, X_1, X_2$ Is there some relation between $Y $ and $Z$ are conditionally independent given $(X_1, X_2)$ $Y $ and $Z$ are conditionally independent given $X_1$, and ...
1
vote
2answers
70 views

Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
1
vote
2answers
245 views

Expected number of steps between states in a Markov Chain

Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{3} ...
1
vote
1answer
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) ...
1
vote
1answer
296 views

$\sigma$-algebra generated by $\pi$-system

Here is a problem from probability with martingales. I want to a better way of writing this than my waffle: Let $Y$ be a random variable and $\pi (\mathbb{R})$ is a $\pi$-system generating the Borel ...
1
vote
0answers
159 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 ...
1
vote
1answer
167 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 ...