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
16 views

proving a statement based on probability theory

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
1
vote
2answers
22 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
0
votes
0answers
10 views

$\overline{X} \rightarrow c$ in probability $\Rightarrow g(\overline{X}) \rightarrow g(c)$ in probability

Prove that if $\overline{X} \rightarrow c$ in probability and if g is a continuous function, then $g(\overline{X}) \rightarrow g(c)$ in probability. Once I think in the situation $|g|<M$, it is ...
0
votes
0answers
9 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
0
votes
2answers
16 views

Can't find intersection of two probabilities.

I have the following problem: While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation ...
0
votes
0answers
20 views

Is this Markov chain irreducible?

Let $(X_n)_{n \in \mathbb{N}}$ be a Markov chain with state space $I = \{0,1\}^m$ and transition probabilities $$p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 & ...
1
vote
0answers
18 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
0
votes
0answers
10 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
0
votes
2answers
24 views

Determing a transition probability matrix

I need some support with this homework exercise: An urn contains at most $N$ balls. Let $X_n$ be the number of balls in the urn after the $n$-th execution of the following procedure: If the urn is not ...
0
votes
1answer
16 views

independence and characteristic functions [duplicate]

Why is it that \begin{equation*} \mathbf{E} [e^{i t_1 X_1} e^{i t_2 X_2}] =\mathbf{E} [e^{i t_1 X_1}]\mathbf{E} [e^{i t_2 X_2}] \end{equation*} for RVs $X_1, X_2$ and all $t_1, t_2\in\mathbb{R}$ ...
1
vote
0answers
17 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
1
vote
1answer
16 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
-4
votes
1answer
20 views

Four letters A B C D are arranged in a line. What is the probability that A and B will always be together [on hold]

Four letters A B C D are arranged in a line. What is the probability that A and B will always be together. Probability Question
0
votes
1answer
12 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
1
vote
1answer
30 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
3
votes
2answers
61 views

What are the odds of any role of a 24 sided die occurring 4 or more times in 10 rolls?

Note that I am not asking about the odds of a chosen roll happening 4 times in 10 rolls, (this has a probability of 0.000517 according to a binomial calculator), rather, the odds of ANY roll happening ...
0
votes
2answers
27 views

Two-dimensional Brownian motion

Let $B_1$ and $B_2$ be two $\mathbb{R}$-valued Brownian motions with $$\langle B_1,B_2\rangle=\int_0^t\rho_s ds,$$ where $\rho$ is progressively measurable with values in $(-1,1)$. We define ...
1
vote
1answer
26 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
0
votes
0answers
17 views

Proving that each element in reservoir have equal probability of been selected in reservoir sampling?

Here is the description of the algorithm and proof of the correctness The algorithm creates a "reservoir" array of size $k$ and populates it with the first $k$ items of $S$. It then iterates through ...
-1
votes
1answer
26 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
1
vote
0answers
11 views

Understanding the difference between convergence in distribution and convergence almost surely

I know that the sum of $\sum_{i=0}^nZ_i$ where $Z\sim N(0,1)$ has a distribution of a Chi squared distribution with $n$ degrees of freedom which in my understand means that $Z^2$ converges in ...
0
votes
1answer
19 views

A die is thrown $n$ times. $X_1$-number of times a number from $\{1,2,3\}$…

.. $X_2$ number of numbers that fell from $\{4,5\}$, $X_3$ number of $6's$ that fell. Find $$P\{ X_1=k\mid X_2=m\};0\leq m \leq n.$$ Now, I believe that $X_3$ is completely irrelevant here. What I ...
0
votes
2answers
21 views

In a box which has balls numbered 1..100 , 5 balls are drawn.

$X$- random variable that represents the largest number of the 5 drawn. Find the distribution of $X$. Now, it seems that this random variable is of discrete type. What I have trouble it defining it ...
0
votes
0answers
19 views

Questions on Kolmogorov Zero-One Law Proof

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Rosenthal's Probability book: Here are my questions: Question 1: In the first red box, does the fact that Q ...
1
vote
1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
0
votes
0answers
25 views

Toss a coin infinitely. Show limiting fraction and 3 consecutive tosses are the same are in tail field

Consider tossing a fair coin infinitely. Let $H_n$ be the event that the nth toss turns up heads. Let $\tau = \bigcap_{n\geq1} \sigma(H_n, H_{n+1}, ...)$. Question 1: Consider the event $A = ...
0
votes
0answers
12 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
2
votes
1answer
24 views

Convergence of probability for $t$-distribution

Assume that $Z_0,Z_1,Z_2,\dots$ are i.i.d. RVs, $Z_j\sim N(0,1)$, and set $$T_n:=\frac{Z_0}{\sqrt{\frac1n(Z_1^2+\cdots+Z_n^2)}}$$ (a) Compute the limit $$\lim_{n\to\infty}\text{P}(T_n^2+2T_n\leq ...
1
vote
0answers
28 views

Square of a weakly stationary process

I have to prove that if $X_t$ is a weakly stationary process, $X_t^2$ is also. It is easy to prove the part referred to the means but I do not know how to work with covariances. Thanks!
-1
votes
1answer
31 views

Markov chain period

Let a Markov chain with State space $E=\{1,2,3,4\}$ and probability transition matrix: $$P=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 1/4 & 0 & 1/4 & 1/2\\0 & 1& 0 & 0 \\ ...
0
votes
1answer
33 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
1
vote
0answers
25 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
0
votes
0answers
6 views

Maximize the net profit with probabilities — optimal purchasing

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
3
votes
1answer
34 views

Is it true that $\sum_{n=1}^\infty \dfrac{Var(Y_n)}{n}<\infty$?

Suppose that $\{X_n\}$ is an i.i.d. sequence of random variables with $E|X_1|<\infty$.Define $Y_n=X_nI_{\{|X_n|<n\}}$ for all $n\geq1$. Is it true that $\sum_{n=1}^\infty ...
-1
votes
0answers
109 views

Pareto distribution,fisher information, confidence interval

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
1
vote
0answers
37 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
1
vote
1answer
22 views

Using characteristic functions to establish convergence

So I have found the Characteristic function of the variable $X_ \lambda$ to be: $$\psi_{X_\lambda}(t) = \psi_{b(\lambda)(Y_\lambda-\lambda)}(t)=\mathbf Ee^{itb(\lambda)(Y_\lambda-\lambda)}=\mathbf ...
2
votes
0answers
24 views

Martingale, increasing sequence of random variables

I want to prove the following: Let $\{X_n\} _{n \in \mathbb{N}}$ be a sequence of random variables such that $P( X_{n+1} \ge X_n)=1$. Then $$\{X_n\} _{n \in \mathbb{N}} \ \text{is a martingale} ...
1
vote
1answer
28 views

Proof of (a step in the proof of) the Law of Large Numbers

Theorem: Let $f:[0,1] \to \mathbb R$ be a measurable function bounded by $c$. Let $U_1,U_2,\ldots,U_n$ be i.i.d. and Uniform$(0,1)$. Then: $$ P \left( \left\lvert ...
1
vote
0answers
13 views

Showing that a sequence of random variables has CLP.

This is an exercise that I am stuck at. I managed to solve (i) and (ii), which are relatively easy. Here the Feller condition in (i) is as below. Also, the central limit theorem I learned is like ...
1
vote
1answer
16 views

Show that $X_n/n$ does not converge almost surely

I am generally able to prove that a sequence of random variables $X_n$ converge almost surely to a random variable $X$ by using the following strategy: Take any typical sample point ...
3
votes
0answers
25 views

Interpretation of composite of random variable

Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$. How do I interpret the following statement: $$ Var(f \circ ...
3
votes
1answer
31 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
0
votes
1answer
41 views

Limit of random walk on $\mathbb{Z}$

$$\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}}=+\infty, \quad\liminf_{n \to \infty} \frac{S_n}{\sqrt{n}}=-\infty \quad P\text{-a.s.}$$ Here $S_n$ is a random walk on $\mathbb{Z}$. I managed to show ...
1
vote
1answer
46 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
2
votes
1answer
23 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
5
votes
0answers
42 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
1
vote
0answers
8 views

multivariate convergence in distribution for Skorohod topology

If $X_n$ is a $R^d$-valued sequence of stochastic processes and $X$ is a $R^d$-valued the limiting process, all having Cadlag paths. I know what it means by $X_n \Rightarrow X$, i.e. $X_n$ converges ...
1
vote
1answer
36 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
1
vote
1answer
27 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...