# Tagged Questions

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### 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 ...
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### 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  ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### $\prod_{n=1}^{\infty} (1- P(A_n))=0$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $P(limsup A_n)=1$.