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
13 views

Prove convergence in distribution using the CLT

Problem. Let $X_1,X_2,...$ be independent and identically distributed random variables such that $EX_1^4<\infty$ and $0<V(X_1)=\sigma^2$. Put $$T_n = \frac{1}{n}\sum_{j=1}^n ...
0
votes
0answers
14 views

Expected number of button clicks

Suppose we have $N$ buttons and each button can be clicked with probability $p_i$. The game stops when the player clicks the button with $i = 1$. What is the expected number of clicks? I am not able ...
-1
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4answers
39 views

If $P(A|B\cap C) = P(A|C)$ then $P(A\cap B|C) = P(A|C)P(B|C)$ [on hold]

Can anybody prove $P(A\cap B|C) = P(A|C)P(B|C)$? I got it up to here. $P(A\cap B|C)= P((A\cap B)\cap C) /P(C)$. Can somebody continue it please or start it from the beginning? Update: It is also ...
0
votes
1answer
28 views

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$?

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$? In a proof I doubtfully used this equation. Is it correct? But I am not sure about it. Can somebody confirm its validity? If possible, can ...
3
votes
1answer
16 views

bounding the measure of a disjoint union under a product measure

Let $(X\times Y,\mathcal{F},P)$ be a probability space such that $P$ is a product measure on $X\times Y$. Let $E=\cup_{x\in X}E_x$ be an event such that: (i) for each $x\in X$, $E_x$ is an event of ...
1
vote
2answers
21 views

If $1\leq \alpha$ show show that the gamma density has a maximum at $\frac{\alpha - 1}{\lambda} $

So using this form of the gamma density function: $$g(t) = \frac{\lambda^{\alpha}t^{\alpha -1}e^{-\lambda t}}{\int_0^\infty t^{\alpha - 1} e^{-t} \, dt} $$ I would like to maximize this. Now i was ...
-1
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0answers
19 views

Conditional Probability on a Mixed RV

TLDR: How do you show $P(Y\in\Omega|X=x_0)=1$ when $Y$ is a mixed RV. Let $(X,Y)\to\mathbb{R}^{dx+dy}$ be a random vector distributed according to a density $f_{X,Y} (x,y)$. Let $X$ be a continuous ...
2
votes
1answer
36 views

Proof that absolute value of a random variable is a random variable

Is this proof correct?: Proof: Suppose that $X$ is a random variable on a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$. Suppose $x \in \mathbb{R}$ and $x \geq 0$. Then $\{|X| \leq x\} = ...
1
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0answers
44 views

Number of rewards before death

I have a question regarding Poisson events with death. Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with ...
1
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1answer
23 views

Conditional expectation of iid nonnegative random variables

I am studying Ross's book, stochastic processes. There is the following lemma: Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then, $E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] = ...
-2
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1answer
13 views

Question about joint convergence with covariance

Just a short question about joint convergence. Assume random variables $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ and the following convergences in distribution $X_n \to X$, $Y_n \to Y$. Furthermore, ...
0
votes
1answer
28 views

If $w$ is a discrete random variable then is $P(w|x)$ a density or mass function? [on hold]

$w$ is a discrete random variable. $x$ is a continuous random variable. Then should I denote $P(w|x)$ as a probability density function or probability mass function, and why ?
0
votes
1answer
11 views

Can a finite set support a $\sigma$-algebra

Assume the set $\Omega$ is finite (finite number of elements). $A$ is a collection of subsets of $\Omega$. It is clear that $A$ can be an algebra, but is $A$ then also automatically a ...
1
vote
1answer
50 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
1
vote
0answers
14 views

Source coding with 2 distinct distributions and entropies

I'm learning about source coding, and many of the books/resources I've read give examples of the source $X^n$ being defined as a sequence of iid random variables. How about when the sequence is ...
6
votes
1answer
37 views

Continuity of a stochastic process

Exercise 3.11 in Oksendal's book "SDEs: an intorduction with applications". Let $W_t$ be a stochastic process satisfying 1) ${W_t}$ is stationary; 2) $t_1\not=t_2\implies W_{t_1}$ and $W_{t_2}$ are ...
2
votes
2answers
29 views

Borel $\sigma$-algebra, definition

This following statement is from a book: Let $C$ denote all open intervals. Since every open set in $\mathbb{R}$ is the countable union of open intervals, we have $\sigma(C)=$ the Borel ...
0
votes
1answer
20 views

Monotone likelihood property and first order stochastic dominance

I have a question regarding first order stochastic dominance. Give two pdf $f(x)$, $g(x)$, $x\in[x_0,x_1]$. For all $x$ on the support, I have $$ g(x) = f(x)\cdot H(x) $$ where $H(x)$ is continuous, ...
0
votes
1answer
26 views

Example of uncountable union of sets that is in $\sigma$-algebra but does not satisfy additivity property of probability

Let $(\Omega, \mathcal{F}, \mathrm{P})$ be a probability space. I know that $\mathrm{P}$ should satisfy: $$ \mathrm{P}(\cup_{n \geq 1}A_n) = \sum_{n \geq 1} \mathrm{P}(A_n) $$ for disjoint family of ...
0
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0answers
32 views

The Gambler's Ruin without using random walks

This is more of a doubt. I understand that this problem can be described with Markov chains and the recursion solved without much trouble. However I've seen that some people casually say that $$ ...
0
votes
2answers
44 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...
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1answer
37 views

What is a nice, clean proof to show that a fair coin toss satisfies axioms of probability?

If we assume H=Heads T=Tails and we're dealing with a fair coin what is a good way we can show that Kolmogorov Axiom has been satisfied?
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0answers
23 views

What can be said about $E(X|N)$ when $N \subset \sigma(X) $?

When $\sigma(X) \subset N $, $E(X|N) = X$ a.e.. When $N \subset \sigma(X) $, is there some result about $E(X|N)$? Or is it no more particular than an arbitrary $N$? Thanks.
1
vote
1answer
24 views

Determine measurability of E(X|N) or even $\sigma(E(X|N))$?

Suppose $(\Omega, F, P)$ is a sample space, $X$ a random variable, and $N$ a sub sigma algebra of $F$. How can we determine $\sigma(E(X|N))$? How is $\sigma(E(X|N))$ related to $\sigma(X)$ and ...
0
votes
2answers
29 views

Inequality for the expected value of the sum of Bernoulli random variables

I'm stuck with this seemingly simple inequality. Suppose that $X_1,X_2,\ldots$ are Bernoulli random variables and denote $S_n=X_1+\ldots+X_n$. Let $n_k=\inf\{n:\operatorname ES_n\ge k^2\}$ for ...
-1
votes
1answer
22 views

Are the converses of the following special cases of conditional expectation also true?

Let $X$ be a random variable, and $N$ be a sub sigma algebra of the underlyign sigma algebra of the sample space. if $X$ is in $L^1$ and measurable wrt $N$, then $E(X|N)=X$ a.e.. Is it true that ...
1
vote
1answer
299 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
2
votes
1answer
325 views

Explicit examples of smooth entropy computation

Smooth classic entropies generalize the standard notions of entropy. This smoothing stands for a minimization/maximization over all events $\Omega$ such that $p(\Omega)\geq 1-\varepsilon$ for a given ...
8
votes
3answers
904 views

What does the -log[P(X)] mean in the calculation of entropy?

The entropy (self information) of a discrete random variable X is calculated as: $$ H(x)=E(-log[P(X)]) $$ What does the -log[P(X)] mean? It seems to be something like ""the self information of each ...
2
votes
0answers
32 views
+50

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
0
votes
3answers
32 views

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart?

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart? There are 52 cards in the deck and there is no replacement. $$P(4\text{th heart} | ...
3
votes
1answer
40 views

$\sum_{n=1}^{\infty}P\left( |X_n-X|>\epsilon\right)<\infty$ for each $\epsilon>0$ $\implies$ $X_n\to X$ a.s.

Problem. Let $\left( \Omega, \mathcal A, P \right)$ be a probability space and $X,X_1,X_2,...$ random variables on $\Omega$. If $\sum_{n=1}^{\infty}P\left( |X_n-X|>\epsilon\right)<\infty$ for ...
1
vote
1answer
31 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
4
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0answers
75 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 ...
0
votes
1answer
32 views

A Characterization of the Strong Markov Property

I have a question concerning the strong Markov property: For a strong Markov process $(X_u)_{u\ge 0}$, a real time $t\in \mathbb{R}$ and an optional stopping time $T$ with $t< T$ \begin{align*} ...
1
vote
3answers
195 views

Find $P(X<Y<Z)$ of exponential density functions

Let X, Y, Z be independent continuous random variables with exponential density functions $\lambda e^{-\lambda x}$, $\mu e^{-\mu y }$ and $\nu e^{-\nu x}$ respectively, on $[0,\infty)$ (and zero ...
0
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0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
11
votes
1answer
82 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
1
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0answers
14 views

Infinite sets of rvs equal in distribution

Assume $\{\mathcal{L}(X_{k})\}_{k\in I}=\{\mathcal{L}(Y_{k})\}_{k\in I}$ for all finite $I\subset \mathbb{N}$ i.e. equal in all finite dimensional distributions. Then I want to show that ...
1
vote
2answers
29 views

Equivalence of Definitions of lim inf of Sequence of Sets

Prove : $\{w : w \in A_n \text{ for all $n$ except a finite number}\}= \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k$. I am trying to prove these two definitions are equivalent but I am having ...
0
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0answers
30 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
2
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0answers
47 views

Transformation of probability density function

I'd like to compute the pdf of $w= g_1(x) = \frac{x}{1+e^{-x}}$ in dependence of the density $f_x(x)$ with domain $x>0$. As I was not able to write the inverse function of $g_1(x)$, I tried the ...
0
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0answers
35 views

What is the probability that from 23 people 2 people have their birthday on the same day?

What is the probability that from 23 at least people 2 people have their birthday on the same day. Assume that the year has 365 days and that all the birthday combinations have the same probability. ...
1
vote
1answer
24 views

Quick question on why two measures have equal total mass.

I am following Probability with Martingales by Williams I am having troubles with why the two measures $H \rightarrow P(I \cap H)$ and $H \rightarrow P(I)P(H)$ have the same total mass $P(I)$. Is ...
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0answers
23 views

how to determine presence of an event with a degree of confidence proportional to a set of observations and conditional probabilities

My probability theory has become a bit rusty and i can't seem to figure out how to determine the presence of a malfunction within a device given a set of observations displaying a certain phenomenon ...
3
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1answer
45 views

Good problem book for convergence concepts in probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
0
votes
0answers
25 views

Markov Property Definition

Let $(X_t)$ be a stochastic process on $(\Omega, \mathcal F, \{\mathcal F_t\}, \mathbb P)$. The typical definition of the Markov property is $\mathbf{P}(X_{t+s} \le x \, |\, \mathcal F_t) = ...
0
votes
1answer
21 views

Show that f is a density and find the corresponding cdf

$f(x) = \frac{(1+\alpha x)}{2} $ for $-1 \leq x \leq 1$ and $f(x) = 0$ otherwise, where $-1\leq \alpha \leq 1$. Show that $f$ is a density and find the cdf. I am mainly having trouble with finding ...
0
votes
1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
0
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0answers
31 views

Example where probability theory fails without $\sigma$-algebra

I have just started reading theory of probability in a measured theory based approach and was wondering if someone could give an example where probability fails without using $\sigma$-algebra (or ...