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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1answer
18 views

Solving for form of CDF that satisfies $G^t(x) = G(t^{-\theta} x)$

For $\theta >0$, I want to solve for the CDF, $G$, that satisfies: $$ G^t(x) = G(t^{-\theta} x) $$ The solution given in my notes and states that it's easy to check that $G(x) = ...
0
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1answer
40 views

Questions on the Weierstrass approximation theorem

From Williams' Probability w/ Martingales: (Ignore red boxed part) Red box: Is that because f is continuous and defined on $[0,1]$? Is there a name for such fact (cont and defined on ...
6
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2answers
177 views

Measuring $\pi$ by throwing darts

I want to give an approximation of $\pi$ in this way: I inscribe a circle in a square then I throw darts at random on the square from far away. If the darts falling on the square are $n$ and the ...
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1answer
61 views

Find pdf of sum of n indp exp RVs w/o using MGFs

From Williams' Probability w/ Martingales: Re $E[f(S_n)]$, how do I obtain $f_{S_n}(s)$? It seems that $$f_{S_n}(s) = \frac{s^{n-1} e^{-\lambda s} \lambda^n}{(n-1)!}$$ I tried computing ...
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2answers
100 views

Questions on Inverting Laplace transforms and Probability

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = ...
3
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1answer
63 views

Brownian motion martingale

I have been wondering about the following equality in the textbook by Liggett. I put a red circle at the position where my question is. They use the theorem that $B_t^2-t$ is a martingale and the ...
2
votes
3answers
122 views

A good book for Probability?

I am doing my research in Functional Analysis, especially in "Generalized inverse of Linear Maps". I have come across Probability by studying only the methods or Distributions(like Binomial, poisson, ...
3
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1answer
200 views

Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$, let $A \in \mathscr{F}$. Suppose $$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | ...
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0answers
9 views

Finding the distribution of the Sample mean of normal distribution (check my working?)

If $X_1,X_2,...,X_n$ are iid $N(\mu, \sigma^2)$, then how do we find the distribution of the sample mean $\bar X = (1/n) \sum_{i=1} ^n X_i$ ? I tried this: $ \bar X \sim N(\sum_{i=1}^n \mu_i/n, ...
3
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0answers
57 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = ...
1
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1answer
39 views

Symmetrically distributed random variables

Let $\{X_n\}$ be a sequence of iid random variables, and let $\{Y_n\}$ be another sequence of iid random variables. Suppose that $P(X_n \leq c) = P(Y_n \geq -c)$ for every real number $c$. Let $S_n ...
6
votes
1answer
114 views

Is a “diagonal-like” set always a null set?

For this question, let $\mu$ be the Lebesgue measure on $[0,1]$ and $\mu^2$ the product measure on $[0,1]^2$. Suppose I have a measurable function $f\colon [0,1]^2\to \mathbb{R}$. For all $r\in ...
2
votes
1answer
29 views

Jensen's and Logarithm / Measure Theory

I have this problem of measure theory dat says: Let $(X,F,\mu)$ be a probability space and $f\in L^1 , f\geq 0$ Prove that $\int_X \log f d\mu \leq \log (\int_X fd\mu)$ And prove that the equality ...
3
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1answer
38 views

Does fractional part converge in distribution to a uniform random variable?

Let $X$ be a continuous random variable with a density function $f(x).$ Let $\{x\}$ denote the fractional part of a real number. I am tryng to prove that $$ \mathbb{P}[\{nX\}\leq z] \rightarrow z, \ ...
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1answer
22 views

Finding the expectation and random variable of a story

Question: A bus runs from point A to point B and it only stops if a passenger needs to get off. The bus started with 20 passengers and to the drivers knowledge they were equally likely to get off at ...
0
votes
1answer
39 views

$P(\sum n X_{n}>m)\leq p_{m}\to 0$ for independent $X_n\sim \mathrm{Bern}(\frac{1}{n^2})$

For independent $X_n \sim \mathrm{Bern}(\frac{1}{n^2})$ the question is to find $p_m$, where m is natural, so that $P\left(\sum^\infty n X_n>m\right)\leq p_m$ and $\lim_{m\to \infty}p_m=0$. Such ...
5
votes
1answer
68 views

Can the integral of Brownian motion be expressed as a function of Brownian motion and time?

Let $W_t$ be standard Brownian motion, and define $$ X_t := \int_0^t W_s ~\textrm{d}s. $$ The marginal distributions of $X_t$ are easy to write down (see here), but it doesn't seem possible to express ...
1
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0answers
25 views

Limsup of a sequence of random variables, equivalences

Suppose $X_i$'s are i.i.d, with the density distribution $f(x) = e^{-x}$, $x \geq 0$. I want to show that $$P(\limsup \frac{X_n}{\log{n}} =1)=1$$ However, I don't know very well how to work with that ...
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0answers
7 views

Confidence bound over sets of linear functions

Let $A$ and $B$ be real-valued random variables (bounded if necessary, but not necessarily positive) and $\delta \in [0,1]$. I am given $n$ samples of each: $A_1,A_2,...,A_n$ and $B_1, B_2, ... B_n$. ...
0
votes
0answers
13 views

correlation test

I have ONE draw from a multivariate normal distribution and I want to see/test if they are independent. In particular let $X=(X_1,X_2,...,X_n)' \sim N(0_{n \times 1},\Sigma_{n \times n})$. I want to ...
0
votes
0answers
14 views

Getting the probability of the difference of a sample mean and a population mean

I have a random variable with a normal distribution $X\sim(0,1)$ of which i have a sample size of $n=100$ and a standard deviation of $\sigma=5$. How can i calculate the probability that the ...
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0answers
22 views

Almost sure convergence of random variable vecctors.

Let $(X_n,Y_n)_{n\geq 1}$ be a vector of random variables such that $(X_n,Y_n) \to (X,Y)$ almost sure. This means that $$\Bbb{P}(\{\omega \in \Omega:(X_n(\omega),Y_n(\omega)) \to ...
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2answers
34 views

Proving An Asymptotic Relation Needed to Answer a Probability Problem

I am working on understanding problem 3.4.13 from Durrett's Probability Theory and Examples. I found a solution online that relies on the following relationship: $$ \sum\limits_{j=1}^n j^{2-\beta}\sim ...
4
votes
1answer
97 views

Stopping times and hitting times for càdlàg processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable ...
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0answers
31 views

Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
2
votes
0answers
50 views

Is an event space naturally a sigma-algebra?

My understandings. I learnt that in mathematical logic, which is the formal-foundation of all mathematical theories, a structure is defined as an ordered pair $(X,\sigma)$ where $X$ is a set and ...
1
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1answer
128 views

Product of numbers is even , when an unbiased die rolled?

An unbiased die is thrown $n$ times. The probability that the product of numbers would be even is $1/(2n)$ $1/[(6n)!]$ $1−6^{−n}$ $6^{−n}$ None of the above. My attempt : we have $3$ even ...
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0answers
15 views

Compute $E(Y_t^2)$ with $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$

Consider the process, $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$. To compute the variance of this process, I need to compute $E[(\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} ...
1
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0answers
11 views

Relationship between independence of random variables and their distributions

Let $\{X_n\}$ be a sequence of random variables on some probability space. In my reading, I encountered that one common assumption on the distributions of $\{X_n\}$ is to assume that $\{X_n\}$ is ...
1
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2answers
43 views

Show that $Y$ and $Z$ are independent and find their distributions

Suppose that $X\sim\exp(\lambda=1)$. Let $Y$ be the integer part and $Z$ be the fractional part. Show that $Y$ and $Z$ are independent and find their distributions. This one is kinda confusing. Any ...
3
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2answers
76 views

Show that every finite family of random variables is tight.

A family of random variables $(X_i)_{i \in \mathcal{I}}$ is said to be $tight$ if for every $\epsilon>0$ there exist a compact set $K_\epsilon$ such that $$\displaystyle\sup_{i \in ...
2
votes
0answers
21 views

Calculate the Expected Return in the Limit as $n \to \infty$

Let's say I have some hypothetical investment which can earn me some return $i$, where $i$ is a random variable. After $n$ periods, my expected return on my investment is: $$ ...
1
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1answer
20 views

For what value of $k$ is $f(x)=\frac{k}{(1+x^3)}$a distrib. function and what is its variance?

Let $X$ be a random variable with the folowing distribution function: $f(x)=\frac{k}{(1+x^3)}$ for all $x>0$. Find a value for the constant $k$ for which $f$ will be a distribution ...
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0answers
77 views

Wald's Identity for non-i.i.d. Case

I am looking for Wald's Identity for non-i.i.d. case as discussed in the following links: https://en.wikipedia.org/wiki/Wald%27s_equation What are the assumptions for applying Wald's equation ...
1
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0answers
17 views

Prove that mutual information between integer and fractional parts goes to zero

For a random variable $X$ with a density function $f(x),$ I want to prove that the following holds: $$ \lim_{n \rightarrow \infty}I(\lfloor nX\rfloor;\{nX\})=0 $$ where $\lfloor x \rfloor, \{x\}$ ...
0
votes
1answer
62 views

convergence of random variable in $L^2$

What would be a pragmatic method to inevstigate the convergence in $L^2$ of a sequence of random variables $X_n$ defined on $\Omega=[0,1]$ by $X_n(\omega)=\omega n$, with Lebesgue measure on $\Omega$? ...
1
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1answer
32 views

What is the distribution of the sqrt of a random variable with exponential distribution

Let $X$ be a random variable with an exponential distribution where: $f(x) = e^{-x}$ for all $x \geq 0$ (i.e. $X \sim \textrm{Exp}(\lambda = 1)$). Calculate >the distribution of $Y = ...
1
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0answers
33 views

If $G(s)=G(p+qs)e^{\lambda(s-1)}$ then $G(s)=e^{\lambda(s-1)/p}$

If $G_{n+1}=G_n=G$ where $G(s)=G(p+qs)e^{\lambda(s-1)}$, how can one conclude that $G(s)=e^{\lambda(s-1)/p}$ the G's above are generating functions, In this exercise one has independent variables ...
2
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0answers
19 views

Reference request for selfdecomposable random variables

In a probability theorem course I'm attending to we are covering stable, infinitely divisible and self-decomposable variables. I am only aware of the formal definitions like: $X$ is self-decomposable ...
1
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1answer
41 views

Distribution of the sum of two Poisson random variables

Let $X_1$ be a random variable with poisson distribution $\text{Poisson}(\lambda_1)$ (i.e. $f(x)=\frac{\lambda^x}{x!}e^{-\lambda}$ if $x \in \{ 0,1,2,3,\ldots\}$ and $0$ otherwise) and let $X_2$ be ...
0
votes
1answer
36 views

convergence in probability to a constant

I have a sequence of random variables $X_n$ defined on $\Omega =[0,1]$ by $X_n(\omega)=\omega n$. Let $\mathbb{P}$ be the Lebesgue measure on $[0,1]$. Does $X_n \rightarrow 0$ in probability? I know ...
2
votes
2answers
45 views

Let X, Y be independent RVs. Calculate the following probabilities

Let $X, Y$ be independent random variables with positive integers values, with distribution $$ P(X=i)=P(Y =i)= \frac{1}{2^i},i∈N^∗$$ Find the following probabilities. (i) $P(\max(X, Y ) ≥ i)$ (ii) ...
0
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0answers
18 views

Moivre's Laplace Formula - malfunctioning

While training and solving question on probability, I came over with the question below. I have tryed to find an information online but did not find any proper information about Moivre's Laplace ...
0
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1answer
22 views

Probabilities involving functions of random variables

I have been looking for a resource to explain this and have had no luck. How does one prove the following statement: Let $u>0$, $X$ a random variable, and let $E(X)$ be the expected value of $X$. ...
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1answer
32 views

How do I know when I need to use Bayes' Theorem?

As I stated on the title.. I have a midterm coming up so probability will be covered on the test so I am wondering when do I know I need to use PHP or Bayes' Theorem? Thank you.
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2answers
30 views

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $DX=E(D(X|Y))+D(E(X|Y))$

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $$DX=E(D(X|Y))+D(E(X|Y))$$ $(D - \text{ variance }E- \text{ expectation })$ This semestar, we have been ...
0
votes
1answer
87 views

Average number of students who will get hw back. [duplicate]

The homeworks of $20$ students is collected, randomly shuffled and returned to the students. What is the average number of students who will get back their own homework? PS: I tried doing this using ...
1
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1answer
41 views

Expected value of sequence obtained by multiplying independent RVs by square-integrable RV

the problem in question is: Let $\{X_n\mid n\ge 1\}$ and $Y$ be random variables on probability space $(\Omega, \mathcal F, \mathbb P)$. Suppose that the $X_n$ are independent, $E[X_n] = 0$ and ...
2
votes
1answer
18 views

Density of $(\frac{Y_{i}}{\sqrt{\sum _{i}^{n}Y^{2}_{i}} })_{i}^{n}$ for iid $Y_{i}\in N(0,1)$ (Computing a determinant)

The density of $\left(\frac{Y_{i}}{\sqrt{\sum _{i}^{n}Y^{2}_{i}} }\right)_{i}^{n}$ for iid $Y_{i}\in N(0,1)$ can be computed by change of density formula. The answer will be rotationally symmetric. ...
0
votes
1answer
22 views

Probability of counter being 23

This is a probability question. Suppose we start our journey from step number 0. We now toss a coin. If the result is a head, we add 3 to a counter. If the result is a tail, we add 2 to the counter. ...