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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2
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1answer
53 views
+50

Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk: A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
0
votes
1answer
42 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
0
votes
0answers
38 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more?

I have made a probability game, where you have to pull out 5 cards (from a deck of 52 cards and two jokers {54 cards total}), and if they add up to 40 or more, they win. Also, if the player pulls out ...
1
vote
1answer
18 views

Notation: the $\sigma$-algebra $\mathcal{F}_\tau^+$

I'm reading a probability textbook on stochastic processes (Jochen Wengenroth's "Wahrscheinlichkeitstheorie", de Gruyter 2008) and the following notation: "$\mathcal{F}_\tau^+$" came up in the ...
0
votes
0answers
28 views

proving a statement on Measure theory

Consider $(\Omega, U, \mu)$ be a measure space and X be an integrable function and for $A$, $\{A_n\}\in \mathscr{U};n\in \Bbb N$ I need to show that $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
0
votes
1answer
24 views

Pareto distribution

So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found ...
0
votes
1answer
61 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
0answers
17 views

Measurability of a set

This question is from Karatzas's Brownian Motion and Stochastic Calculus page 108. Let $W_t$ be standard one-dimensional Brownian motion then it concludes that the set ...
2
votes
1answer
27 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
0answers
17 views

Question concerning invariant distribution

Let us consider the Markov chain $(X_n)_{n \in \mathbb{N}}$ 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
16 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
5
votes
1answer
96 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
0
votes
0answers
8 views

probability generating function moments for the multivariate case

Suppose ${\bf X} = (X_1, \ldots, X_d)$ is a non-negative integer-valued random vector, with pmf $p$, I tried to extend the results of the univariate generating function to the multivariate case, is ...
-2
votes
0answers
191 views

Pareto distribution,fisher information, confidence interval [on hold]

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 ...
0
votes
2answers
31 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
1
vote
1answer
25 views

probability of 26 letters

A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random. A. What is the probanility that the word HAMLET appears somewhere in the string if letters? ...
1
vote
0answers
50 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
votes
2answers
31 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
0
votes
0answers
9 views

Poisson Distribution Optimization Problem

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 ...
2
votes
3answers
23 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
1
vote
2answers
27 views

Conditional probability with students seating

There are 14 students and 9 of them are friends. Students purchased tickets to movie and they got seats in a row of 14 seats. 8 friends got seats next to each other. What is the probability that ...
0
votes
1answer
21 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 ...
0
votes
1answer
40 views

proving a statement based on probability theory [on hold]

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
votes
2answers
44 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
2answers
20 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 ...
2
votes
0answers
23 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
29 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 ...
1
vote
0answers
18 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
28 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
17 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 $t_1, t_2\in\mathbb{R}$ ...
7
votes
1answer
993 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
1
vote
0answers
23 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 ...
-2
votes
1answer
29 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
1answer
18 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 ...
1
vote
4answers
61 views

A Stochastic Limiting Inequality Proof

Let $(X_p)_{p\ge 0}$ be a sequence of non-negative random variables with finite mean for each $p\ge 0$. Then $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
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 : ...
2
votes
1answer
27 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 ...
5
votes
0answers
43 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 ...
3
votes
2answers
63 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 ...
1
vote
1answer
86 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 ...
2
votes
2answers
147 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and ...
0
votes
2answers
29 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
30 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
vote
0answers
29 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
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
0answers
26 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
2answers
22 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 ...
3
votes
1answer
33 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 ...