Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Why $\{Z \le z\} = \bigcap_{m = 1}^\infty \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty \{ Z_k \le z + 1/m \}$ if $Z=\lim_nZ_n$?

I am following A first look at rigorous probability theory by Rosenthal, and I am having troubles with limits of random variables. Specifically proposition 3.1.5. (iii) states that if $Z_1,Z_2...$ ...
2
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1answer
58 views

If $X$ is Poisson, find the expectation of $\frac{1}{a+X}$

If $X$ is a Poisson random variable with $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$ and $a>0$ then find the expectation of $\frac{1}{a+X}$ If I make use of ...
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0answers
26 views

Native algorithm for lottery

Consider a simple lottery game that you are required to pick 6 numbers out of 50 numbers (1 to 50), and you have the history of the most recent n games' result, by ...
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0answers
25 views

Soft Question: What are some elementary motivations of using functional analysis to study probability theory?

Recently i've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure theoretic ...
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2answers
40 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where ...
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1answer
35 views

Carleman's conditions

I compared Carleman's condition to Hadamard's radius of convergence for Taylor series. Given that the MGF can be re-expressed as a taylor series (that can be extended to a strip in the complex plane ...
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0answers
12 views

Proving that an inductively defined function is a Markov chain

Let $X_0$ be a random variable with values in a countable set $I$. Let $Y_1,Y_2,\ldots$ be a sequence of independent random variables, uniformly distributed on $[0,1]$. Suppose we are given a function ...
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1answer
35 views

Calculate the expected value

To get the expected value of $E(X), E(Y) $ and $E(X, Y)$ given: $$ f_{X,Y}(x,y) = 3x $$ where $0\le x \le y \le 1.$ My solution is, first get the margin distribution: \begin{aligned} f_x(x) &= ...
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0answers
21 views

Meaning of (generalized?) differential operator

I am currently reading this paper which makes use of generalized differential operators. As I understood it, the operator $D_x$ works like this: If $F$ is a continuous function on $[a,b]$ and $G$ an ...
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0answers
29 views

Can anyone give me an intuitive explanation on this statement? [closed]

No two distinct distribution have the same characteristic function.
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1answer
32 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
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1answer
18 views

Expected Residual lifetime

I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
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0answers
23 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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0answers
23 views

Change of measure

Title might be wrong. Anyway I am given that $\delta$ is nonnegative and measurable in $(\Omega, \mathfrak{F}, \mu)$. It can be shown that the mapping $\nu: \mathfrak{F} \to [0, \infty]$ defined by ...
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1answer
22 views

probability of input word

I have 2 text files containing certain sentences. I calculated the individual probabilities of every single word in both files for file1 p(A)=[(total occurrence of a word in file 1)/ (total no of ...
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1answer
16 views

Intersection of subsets of $\sigma$-algebra

Let $\mathscr{F}$ be a $\sigma$-algebra on a countable set $\Omega$.For some $\omega\in\Omega$ define $$S(\omega)=\bigcap\{B\in\mathscr{F}:\omega\in B\} $$ Is $S(\omega)\in \mathscr{F}$?
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13 views

Large-cardinality probability spaces

Given a continuous random variable, the way to a calculate the probability that it will fall within some range of values is to integrate its density function over that range. My question is: is there ...
1
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1answer
22 views

Joint Quadratic variation

Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I ...
0
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1answer
44 views

Conditional PDF of $X_1$ conditionally on $X_1 + X_2 > a$ if $X_1$ and $X_2$ are i.i.d. exponential

Thank you all in advance. I'm trying to calculate: pdf of $(X_1 | X_1 + X_2 > a)$ ($X_1$,$X_2$ ~ exp($\lambda$)) rewriting $$ X_3 = X_1 + X_2$$ then we need to find $$P(X_1 | X_3 > a)$$ I know ...
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4answers
69 views

Chance of playing a game

You are offered a chance to play a game. the rules are simple. there are $100$ cards face down. Of these, $55$ say win and $45$ say lose. You begin with $10000$ dollars. You must bet $1/2$ of your ...
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1answer
44 views

I have got 8 to be in game I have to pay 1 each time. I can win 2. What is a probabilty that I will have 10?

I have got 8 dollars. To be in game I have to pay 1 dollar each time. I can win 2 dollars in every game with probability $\frac{1}{2}$ I have got 10 chances. If I have 0 dollars the game is over. What ...
3
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1answer
29 views

how to find ALL $\sigma$-algebras of a given sample space?

I have a sample space $\Omega$={$\omega_{1}$,$\omega_{2}$,$\omega_{3}$,$\omega_{4}$} and I need to find ALL $\sigma$-algebra on $\Omega$. I know how to construct some $\sigma$-algebra like ...
3
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1answer
47 views

Conditional expectation Doubt

Let $\{X_n\}$ be non-negative integrable random variables on $(\Omega, \mathscr{F}, \mathscr{P})$ adpated to a filtration $\{ \mathscr{F_n}\}$ and bounded a.s. by a constant $C < \infty$. Let $\{ ...
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0answers
21 views

Bayesian Chain rule

I am going thorugh a Naive Bayes Classifier, and faced the following: $p(y|a,b,c) = \frac{p(a|y,b)*p(y|c)}{p(a|b,c)}$ When I am trying to derive the above, these are my steps: ...
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2answers
37 views

Show $\sigma$-algebra of cylindrical sets is a certain collection

I am trying to prove that $B\in$ the $\sigma$-algebra generated by the cylindrical sets of the form $\{f: [0,1] \to \mathbb{R} | (f(t_1),...,f(t_n))\in A\}$ for some $n \geq 1$ and Borel subset ...
2
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1answer
39 views

Convergence of expectation of logarithms of random variables

Let $\{ X_n \}$ be a sequence of random variables that converge to $X$ in distribution. Assume that $P\{X_n \geq 1\} = 1$ for all $n$, and that $\mathbb{E}X_n \rightarrow c < \infty$. Does it ...
2
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1answer
27 views

Sufficient conditions for a sum over a countable set to be well-defined

Suppose $W$ is a countable set and $f:W\to\mathbb{R}$ is a real-valued function. I would like to know the sufficient conditions so that the concept $$ \sum_{w\in W}f(w)\tag{$*$} $$ is well-defined. ...
3
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1answer
13 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
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1answer
39 views

Breaking up a countable sum

Let $W$ be a countable set and $p:W\to[0,1]$ be any function satisfying $$ \sum_{w\in W}p(w)=1. $$ Now, let $A_1,A_2,\ldots$ be a countable collection of disjoint subsets of $W$. With $A=\bigcup_n ...
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1answer
44 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
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1answer
13 views

Geometric Random Variable of a coin toss which is tossed 1 time.

The geometric random variable is defined (in this example) as the number of tosses needed for a head (a fair coin) to come up for the first time. $P_x(k)$ = $(1 - p)^{k-1}$ * $p$ So I calculated the ...
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1answer
23 views

How is this Negative Binomial Random variable used to solve this problem?

I was looking at the solution to this problem below and I don't understand how they used a negative binomial R.V. to solve the problem. A research study is concerned with the side effects of a new ...
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3answers
282 views

n people & n hats: probability that at least 1 person has his own hat

Suppose n people take n hats at random. What is the probability that at least 1 person has his own hat? The proposed solution uses inclusion-exclusion principle and gives the answer: ...
2
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2answers
46 views

Unordered sampling with replacement

I have difficulties with understanding of how do we count this. Why can't we just divide number of ways of ordered sample with replacement $n^k$ by $k!$, i.e. to get rid of permutations since we ...
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0answers
28 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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0answers
21 views

Probability to find Divisibility of an integer by prime numbers [closed]

What is the probability that a positive integer n<100 is divisible by a prime number p<100?
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1answer
17 views

Average Waiting Time for a General Process

The time between the arrival of two consecutively buses are independent and averages out to be $T$. A passenger arrives at a uniformly distributed random time independent of the bus arrival time. Can ...
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1answer
57 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
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3answers
28 views

The limit of the probability that a random variable is greater than a constant is zero?

I was just wondering if $\lim_{a\rightarrow\infty}P(X>a)=0$ for any random variable $X.$ It seems quite obvious but can we prove it?
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1answer
45 views

An inequality involving the sum of the probability of intersection of sets with same probability

Suppose $P(A_i)=p$ for any $1\leq i\leq n.$ Prove that $$n(n-1)p^2-\sum_{j,k;j\neq k}P(A_j\cap A_k)\leq n.$$ Hint: consider Cauchy or Jensen's inequality. My attempt: \begin{eqnarray*} ...
3
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1answer
63 views

X,Y,Z are mutually independent random variables. Is X and Y+Z independent?

X,Y,Z are mutually independent random variables. Is X and Y+Z independent? Please, give me a hint how to prove it?
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1answer
34 views

Proof of extinction probability in Galton-Watson-process using a Martingale

this problem is somewhat similar to the thread The extinction probability of Galton-Watson process from a Martingale perspective. I want to show, that for a Galton-Watson-process $Z_0,Z_1,\ldots$ with ...
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2answers
71 views

Bounding the size of all consecutive sums of i.i.d. random variables

Let $X_1, X_2, \ldots$ by an i.i.d sequence of bounded (for simplicity), mean zero random variables. For any $a<b$, call $S_{a,b} = X_{a+1} + \cdots + X_b$. I would like to show that for any ...
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41 views

How to work with the mode of a probability mass function

How do you work with a probability mass function in determining stuff related to the mode. Here's the question I have $P(X=x) = {\theta^n}{{n}\choose{x}}({\frac{1-\theta}{\theta}})^x, x = ...
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38 views

Bayesian Parameter Estimation - Notation in Terms of Probability Spaces

As far as I know, random variables are functions form a probability space $(\Omega,\mathcal{A},\mu)$ to real numbers $\mathbb{R}$, i.e. $X:\Omega\to\mathbb{R}$. Let a probability density function ...
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1answer
40 views

Jumping times of a Lévy Process

If one has a Levy-process, are the times when the process has a jump of size exceeding a positive $\varepsilon$ actually stopping times w.r.t. the canonical filtration? In more detail: Let ...
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2answers
28 views

Does a $\sigma$-finite measure always admit a countable partition whose components are uniformly bounded from below and/or above?

Let $(A,\mathcal{A})$ be a measure space and $\mu$ be a $\sigma$-finite measure on $A$ such that $\mu(A)=\infty$. Is it true that then one can find a partition $(A_m)_{m \geq 1}$ of $A$ such that ...
3
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2answers
81 views

Proving integrability of a random variable involving stopping times

Let $X_1, X_2,...$ be i.i.d integrable random variables in $\mathbb{R}$ with $\mathbb{E}[X_i] =0$ and $\mathbb{P} (X_i >0) >0$. Let $x>0$, $S_0 = x$, and $S_n= x + \sum_{i=1}^{n} X_i $. For ...
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2answers
69 views

Question on the application of Kolmogorov Zero-One law

Let $X_1$, $X_2$,... be independent random variables on ($\Omega$, $\mathcal{F}$, $\mathbb{P}$). Suppose that the $X_i$ are symmetric (i.e. $X_i$ and $-X_i$ have the same distribution) and that there ...
0
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1answer
24 views

Computing absolute distribution from conditioned probability

for a sum $X:= \sum_{i=1}^n 1_{U_n<U_0}$ of a series of random variables $U_1, ..., U_n$, all of them uniformely distributed on the unit interval as is $U_0$, I computed the following conditional ...