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
48 views

Finding $\mathbb{E}(X+1)$ and $\mathrm{var}(X+1)$ of a Poisson rnd variable

In this exercise: Let $X$ be a Poisson random variable with parameter $\lambda$ and let $Y=X+1$. Find $\mathbb{E}(Y)$ and $\mathrm{var}(Y)$. I was able to apply the definition of expected value ...
2
votes
1answer
78 views

Why can strong law of large numbers be applied in this question?

Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables taking values in the set of natural number $\mathbb{N}$. Assume that $\mathbb{P}(X_1=i)=p_i>0$ for $i\in\mathbb{N}$. Let $D_n$ denote the ...
0
votes
1answer
176 views

When is a random variable is said to be well-defined?

In the paper On the Bootstrap of the Sample Mean in the Infinite Variance Case by Keith Knight, on page 1170 at the bottom of the page before the theorem, the author mentions that the random variable ...
1
vote
1answer
37 views

Sub-additivity in a probability space

I am proving th generalization of sub-additivity in probability. And i cant proceed to the next step because i cant show that this is true: $P( \bigcup A_i - \bigcup B_i )= P( \bigcup A_i) - P( ...
4
votes
1answer
37 views

An application of law of large numbers

How can one apply a law of large numbers to a Poisson Process in order to deduce the analytic fact that $$\lim_{t\rightarrow\infty} e^{-t}\sum_{n=0}^\infty ...
1
vote
1answer
19 views

What is the relation between fix points of random uniform permuation, and probability of independent events occuring.

Let $A_1,\dots,A_n$ be independent events that occur with probability $1/n$ each. Let $p_{n,k}=P($exactly $k$ events occur). One can show with stirlings formula that ...
0
votes
1answer
95 views

HWK Help: Find the Probability that a Family has Exactly k Boys

I've reviewed a similar proposed question, however the help given wasn't exactly what I was looking for (unfortunately). So if the probability that a family will have exactly n children are equally ...
0
votes
1answer
84 views

Toss coin till two heads in a row or two tails in a row. Let N = total tosses and X = N − 1. It takes at least two tosses, Im(X) is 1,2,3,···. Find p.

I understand how to show that X is a geometric random variable, because the image is 1,2,3,... What I was trying to do was to look at the geometric distribution P(X = x) = pq^(k-1) but could not ...
1
vote
1answer
22 views

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...
3
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5answers
524 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 suggest a book (or more than one if necessary) that is: rigorous and accurate ...
0
votes
2answers
31 views

Does it matter here that random variables are jointly normally 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 ...
0
votes
1answer
32 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 ...
4
votes
1answer
114 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 ...
1
vote
1answer
76 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 ...
0
votes
1answer
57 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 ...
2
votes
1answer
73 views

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

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 $E[X_1 \mid \mathscr{G_n}]=E[X_1 \mid S_n]$, and that $E[X_j \mid ...
0
votes
1answer
34 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 $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ ...
1
vote
1answer
52 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 ...
0
votes
1answer
81 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.
0
votes
1answer
80 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 ...
1
vote
1answer
70 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 ...
0
votes
1answer
32 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: ...
2
votes
1answer
148 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, ...
2
votes
1answer
983 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 ...
0
votes
1answer
78 views

Corollary of Kolmogorov zero-one law [duplicate]

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, ...
2
votes
2answers
233 views

Show that $\prod (1- P(A_n))=0$ 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 ...
0
votes
1answer
24 views

Generalization of sub-additivity

**strong text**I dont know how to start or construct on how to prove this. If i have a $(\Omega, \Beta , P)$ as probabilty space. How cn i show that for events $B_i \subset A_i$, this generalization ...
1
vote
1answer
63 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 > ...
1
vote
0answers
37 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 ...
0
votes
1answer
20 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 ...
1
vote
1answer
122 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 ...
2
votes
1answer
89 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 ...
0
votes
2answers
30 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 ...
11
votes
1answer
184 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 ...
1
vote
0answers
78 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 = ...
1
vote
1answer
54 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 ...
0
votes
1answer
30 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 ...
2
votes
1answer
119 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 ...
3
votes
1answer
55 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 ...
1
vote
2answers
70 views

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 ...
4
votes
1answer
115 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 ...
0
votes
1answer
25 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 ...
2
votes
4answers
723 views

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 ...
2
votes
3answers
48 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} ...
1
vote
1answer
67 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 ...
0
votes
1answer
53 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 ...
0
votes
1answer
212 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 ...
2
votes
2answers
59 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 ...
1
vote
1answer
235 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 ...
1
vote
1answer
67 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} & ...