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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A probability distribution has values 0, 1, 2, 3, . . . with probabilities πk = c(e^−λ)λ^(2k)/k! for a certain c. What is c?

I understand that this looks similar to Poisson Distribution, but I am unsure how to apply the fact that the probability distribution has values 0,1,2,3...
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8 views

Suggestion: good book on probability theory with emphasis on applications to other areas of mathematics and physics

On this website, there are many questions about books on probability theory, but I would like to ask if you can select (from all the references available on this website and elsewhere) a book ...
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2answers
11 views

Does it matter here that random variables are jointly distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
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7 views

Rigorous Order Statistics with Indicator Functions

If three people are randomly placed along a 1 mile road, the probability that no two of them are less than $m$ miles apart for $m \leq \frac{1}{2}$ could be solved by using the density for the order ...
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13 views

Show the following definition does not give a $\sigma$-addtive measure pathwisely

Given the space of all square-integral functions over $[0,1]$: $L^2([0,1], \mathcal{B}([0,1]), m)$ and a Brownian motion $W_t$ defined on the probability space $(\Omega, \mathcal{F}, P)$, we define ...
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10 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
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13 views

Probability that 3 of 4 people are on 4 of 12 seats by allowing occupancy of seat for more than on person.

I asked a similar question here which is as follows: Four (identical) persons enter a train (section A has 4 seats, section B has 8 seats). What is the probability that exactly (not more or less) 3 ...
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1answer
20 views

For $X_n$ iid, $S_n=\sum X_n$, $\mathscr{G}_n=\sigma(S_n,S_{n+1}, \dots)$, show $E(X_j|\mathscr{G}_n)=E(X_1|S_n)$

If $(X_n)$ is iid in $L^1$, and $S_n = \sum_{i=1}^{n} X_i$ and $\mathscr{G_n} = \sigma(S_n, S_{n+1},...)$, then show that $E[X_1|\mathscr{G_n}]=E[X_1|S_n]$, and that ...
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1answer
10 views

Showing $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ for Martingale $M_n$.

Let $(M_n)$ be a martingale with $M_n \in L^2$. $S,T$ are bounded stopping times w $S\leq T$. Show $M_T, M_T$ are both in $L^2$ and that $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ ...
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6 views

Limes superior of an ergodic and stationary process

Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that ...
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1answer
9 views

Application of Kolmogorov's Zero-One Law

Most of the books I read, they only state some examples of tail events. One of them is $[\sum_n X_n \ converges]$. My main problem is to show that this is indeed a tail event.
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1answer
22 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
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1answer
21 views

Which random variable distribution can be scaled towards zero mean and unit variance?

can any random variable, not necessarily normally distributed, scaled and shifted in such a way that the new mean is 0 and the new variance is 1? If not, which can? I remember hearing about ...
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22 views

Trouble with Conditional probability and expectation [on hold]

I have a few questions in probability that have been bothering me. The first is this: Why is $$E(T-t | T \ge t) = \int_t^\infty \frac{(s-t)f(s)~ds}{P(T\ge t)}. $$ The second is this: How does one ...
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10 views

How do I describe probabilities of realised variables in terms of CDFs and PDFs?

I have a game theory problem here where the realisation of the random variable $Y₂$ (termed $y₂$ is observed in the first stage of the game. $Y₂$ is the second highest order statistic for $n-1$ ...
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1answer
19 views

Inequality with an integral of probability v.s. summation of probabilty

I was reading a proof in probability text and stuck with one line which confusing me. Suppose $\{X_i\}$ are i.i.d. real-valued random variables sequences with $E|X_i| = \infty$ By applying the fact: ...
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1answer
12 views

Beta/Dirichlet question

A generalization of the beta distribution is the Dirichlet distribution. In its bi-variate version, (X,Y) have pdf $f(x,y) = Cx^{a-1}y^{b-1}(1-x-y)^{c-1}, 0<x<1, 0<y<1, ...
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1answer
26 views

How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
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1answer
26 views

Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
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2answers
89 views

$\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Let $A_n$ be independent events with $P(A_n) \neq 1$. Show that $\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$ It kind of looks obvious but I really have no idea how to prove it. Can ...
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1answer
39 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
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12 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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1answer
10 views

Expressing $P(X \vee Y | Z)$ without the disjunction operator

I'm trying to express $P(X \vee Y | Z)$ without the disjunction operator. I have the following already, but I am not sure whether this is correct. $P(X \vee Y | Z) = \frac{P\left(\left( X \vee Y ...
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1answer
30 views

Change in probability complexity when adding 2 “wildcards” (jokers) to a standard 52 card deck

I am wondering what happens to the complexity of probability when "wildcard" conditions are allowed in random card draws. For example, the probabilities of the $5$ card poker hands from a standard ...
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1answer
41 views

How to calculate conditional probability with inequality

I know that: \begin{equation}\displaystyle P(A=x|A+B=y) = \frac{P(A=x \cap A+B=y)}{P(A+B=y)}\end{equation} Assuming $A$ and $B$ are independent, the intersection of the two events can be resolved as ...
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0answers
29 views

Winning probability calculating in a two-players game

The problem comes from an contest that is already over. There are two players playing the game. Given n cards each containing a number. In one go, any one of them ...
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14 views

Calculate expected value of discrete-time Markov jump linear system

I have the following discrete-time Markov jump linear system (state-space system where the state matrix $A$ can differ in time). $$ x(k+1) = A_{\sigma(k)} x(k) $$ With initial conditions $x(0) = ...
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2answers
17 views

Simple finding the PDF given function

I am a little confused on how to go about finding the PDF given a condition for a function. So I have the function $$ Y(x)=ae^{-bx} \,\,\,\,\,\,\, a,b,x \geq0 $$ and I need to find the value for X ...
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0answers
22 views

Learning roadmap: 'combinatorial' probability

I am about to finish working through Williams's Probability With Martingales. I have studied analysis up to the first five chapters of Folland's text but have not studied any combinatorics yet. It ...
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25 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
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1answer
23 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
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26 views

Number of ways to shuffle a cardset with fixed top 4 while ignoring the suit

I am interested in the number of possible orders in a standard 52 card deck. There are $52!$ possible orders, if I care for suit and type. If I don't care for the suit / color of the card there are ...
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1answer
17 views

Pre-image of Borel sets is closed under complements

Let $X$ be a random variable defined on some sample space $\Omega$. Consider the collection $\mathcal{B}_X = \lbrace X^{-1}(A) : A\in \mathcal B (\mathbb R) \rbrace$, where $\mathcal B (\mathbb R)$ is ...
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Equality for the one-sided chebyshev inequality [on hold]

How to show that for any random variable with mean $\mu$ and variance $\sigma^2$ and any $a>0$, $P(X> \mu + a) = \frac{\sigma^2}{\sigma^2 + a^2}$
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1answer
25 views

Corollary of the Kolmogorov Zero-One Law, proof

Here is an application of the Kolmogorov Zero-One Law given in my textbook (a probability path by Resnick page 107-108). It states that the random variables $\limsup_nX_n$ and $\liminf_nX_n$ are ...
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1answer
42 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
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0answers
14 views

Balls and Bins: Probability that every bin contains at most $O(logn)$ balls

I consider the balls and bins experiment, where we have $m=n\log n$ balls and $n$ bins. Every ball uniformly at random chooses one bin. We want to show that with probability $1-o(1)$ every bin ...
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2answers
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Finding the variance of Linear combinations

Here are two questions of similar style from my past CIE A level exams, Now I am unsure how to find the variance in each case, If X and Y are independent random variables, the variance of ...
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0answers
34 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
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1answer
18 views

Conditions on p, f such that $E[H(\tau)] = pH(0) + \int_0^\infty H(t)f(t)dt$ is an expectation

Suppose that a person has to wait a time t before being seated, and that $$E[H(\tau)] = pH(0) + \int_0^\infty H(t)f(t)dt$$ for all functions $H$ for which this expression is defined. What are the ...
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Fair coin tosses: Probability of $\geq 4$ consecutive heads

I know that there are some related questions, but they seem to be overkill for this small exercise. I have 10 (fair) coin tosses and am interested in the probability that I have at least 4 ...
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16 views

If $(A_n)_{n\in\mathbb{N}}$ is a collection of independent events with $P(A_n)<1$, and $P(\bigcup_n A_n)=1$ then the $P(\limsup A_n)=1$. [on hold]

If $(A_n)_{n\in\mathbb{N}}$ is a collection of independent events with $P(A_n)<1$, and $P(\bigcup_n A_n)=1$ then the $P(limsup A_n)=1$.
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3answers
40 views

How do I show that $Var(Y) = n\frac{(\theta _{1} - \theta _{2})^{2}}{(n+1)^{2} (n+2)}$?

My original pdf is $f(y) = \frac{n (y_{n} - \theta_{1})^{n-1}}{(\theta_{2} - \theta_{1})^{n}}$ for $\theta_{1} < y < \theta_{2}$. After using U-substitution, I obtain $E(Y) = \frac{n \theta_{2} ...
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1answer
35 views

Convergence of the expectation of a non-continuous function

Suppose that $F_{n}$ converges to $F$ weakly, where $F$ is a continuous distribution function. Also, suppose that $g$ is a bounded, continuous function and $\{x_{n}\}$ is a real-valued sequence of ...
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1answer
41 views

HWK Help: Conditional Probability Proof

So I've been working on this proof (and most likely making harder than it is) for quite some time now and I am getting nowhere. The proof is the following: Let $A_{k}$ be the event that the animal ...
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1answer
15 views

Random Binary Search Tree, expected value of nodes with two children

In class, the professor showed that using a uniform random permutation $$ X_1,..., X_n$$ (each being i.i.d.) we can construct a Binary Search Tree by inserting the values in to the tree by their ...
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15 views

Sum of the smallest or greatest k components of a random vector drawn from a symmetric Dirichlet distribution? [on hold]

Is any distribution known for the sum of the smallest or greatest $k$ components of a random vector drawn from a symmetric Dirichlet distribution?
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2answers
29 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
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1answer
32 views

If $Y$ is measurable with respect to $\sigma (X)$ then there is a measurable function $f$ so $f(X)=Y$ - Stuck in Proof.

I have $X$ an $\mathbb{R}^n$ random variable, and $Y$ is $\mathbb{R}$ valued that is measurable with respect to $\sigma (X)$. I'm trying to follow a proof showing that there is a Borel measurable ...
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1answer
26 views

Literature request on Markov Chains which state transition probability matrix evolves over time

I want to know is there any literature on markov chains who's state transition probability matrix evolves over time? For instance, I have 2 states, 1 and 2. With $$P = \begin{bmatrix} p_{11} & ...