0
votes
0answers
12 views

Problem with constructing a uniform probability measure over $\mathbb{N_0}$ using rationals on the unit interval

I've been toying with the possibility of constructing a uniform probability measure over $\mathbb{N^0}$. Obviously, one cannot just assign each non-negative integer a probability of 0 and call it a ...
0
votes
0answers
20 views

statistics problem, where did I mistake?

I searched interesting problem about statistic from http://www.mast.queensu.ca/~stat353/resources/pastfinals/final12sol.pdf $$ $$ But at the question No.2, I have some problem the red box $$ $$ ...
0
votes
2answers
39 views

The water heater problem ( mathematician or plumber)??

Isn't it absurd, I mean doesn't it make probability absurd. $\textbf{Problem-}$ Suppose my water heater broke and heat in my apartment raised high. I went to a "person" to ask him to take a look at ...
0
votes
0answers
17 views

Generalization of the Glivenko-Cantelli Theorem

The classic Glivenko-Cantelli Theorem states that $$ \sup_{t}|F_{n}(t) - F(t)| \longrightarrow_{a.s.} 0 $$ where $F_{n}(t)$ is the empirical cdf. Looking at the proof of the theorem, it seems to me ...
2
votes
2answers
21 views

Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
2
votes
1answer
26 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 ...
0
votes
0answers
26 views

maths probabilities questions

n will not exceed 50. a)Consider a random collection of n individuals in a room. Suppose $P_n$ symbolizes the probability to have at least two individuals with the same birthday(born in the sane ...
1
vote
0answers
20 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 ...
-2
votes
0answers
28 views

Borel function and random variable [on hold]

Positive part of a function X+ is borel function and will be a random variable if X is random variable. I know this things, but how should i prove it mathematically ? It can also be put up as, "Borel ...
0
votes
0answers
22 views

concentration of the following random variable: number of items that fit in

This is related to this previous question. Let us assume that we have a capacity $n>0$ which tends to infinity. We are given an i.i.d. sequence of nonnegative random variables ...
1
vote
1answer
12 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
0answers
14 views

Hypothesis testing CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
0
votes
1answer
38 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
0answers
15 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 ...
0
votes
1answer
18 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 ...
2
votes
0answers
23 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
26 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
27 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 ...
0
votes
1answer
29 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
24 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 ...
-2
votes
0answers
25 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 ...
0
votes
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: ...
2
votes
1answer
13 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
54 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 ...
2
votes
2answers
95 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 ...
1
vote
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 > ...
1
vote
0answers
17 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
11 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
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 ...
2
votes
1answer
43 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
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 ...
0
votes
0answers
18 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) = ...
0
votes
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 ...
1
vote
0answers
35 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
40 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 ...
3
votes
1answer
43 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 ...
0
votes
0answers
24 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 ...
1
vote
2answers
12 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 ...
3
votes
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 ...
0
votes
1answer
19 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
104 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 ...
0
votes
0answers
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 $P(\limsup A_n)=1$. [closed]

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 $P(limsup A_n)=1$.
2
votes
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} ...
1
vote
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 ...
-1
votes
0answers
15 views

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

Is any distribution known for the sum of the smallest or greatest $k$ components of a random vector drawn from a symmetric Dirichlet distribution?
0
votes
1answer
28 views

Probability of receiving $k$ numbers out of $n$ in increasing or decreasing order

Suppose you receive a sequence of $n\in \mathbb{N}^+$ numbers chosen independently and at random from a uniform distribution over the first $n$ natural numbers. What is the probability that, within ...
0
votes
1answer
17 views

How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
1
vote
1answer
26 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
1
vote
1answer
30 views

Random distribution of colored balls into boxes.

This is an abstraction of a real problem I have: I have a large number of balls that are either Red or Blue ($n = 9*10^7$) and a bunch of containers ($c = 3*10^7$). I've calculated that the ...
1
vote
1answer
40 views

What to do when a probability problem becomes unwieldy to check via simulation?

I am assuming that some probability problems cannot be solved easily since there may be a lot of cases to handle and it would make miscounting likely. However, some problems do not simulate well on a ...