This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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1answer
84 views

A detail on a proof of the strong Law of Large Numbers.

In the following blog post https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ one is presented with a nice account of the LLN. Suppose that I have shown that if $(n_j)$ is a ...
2
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1answer
66 views

Expected number of drawings to find collision

Consider a group $G$ consisting of $n$ distinct elements. Suppose we draw random elements of $G$ (one by one, replacing each element every time) until we draw an element that was drawn before (we say ...
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0answers
37 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
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0answers
17 views

An asymptotic ratio of samples

Let $S_n = \left\{X_1, \dots, X_n \right\}$ be a sample of idd random variables for all $n \in \mathbb{N}$. Consider two sequences of measurable sets $\left( A_n \right)$ and $\left(B_n \right)$ such ...
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2answers
111 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
2
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1answer
34 views

How to compute $\mathbb{E}(\prod_{i=1}^n(1+X_i)\textbf{1}_{\prod_{i=1}^n(1+X_i)\leq M})$

I want to compute $\mathbb{E}(\prod_{i=1}^n(1+X_i)\textbf{1}_{\prod_{i=1}^n(1+X_i)\leq M})$, where $\textbf{1}$ is the indicator function and $X_i$ are continuous independent and equidistributed ...
0
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1answer
53 views

Bins and Balls problem several balls at once

I'm trying to calculate the expected value of the number of balls that i need to choose for fill all bins with at least one ball. There are $N$ empty bins labeled from 1 to $N$, and infinitely many ...
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0answers
16 views

Definition of expected value of a continuous random variable [duplicate]

Let $X$ be a random variable with the probability desntiy function $f$. Then, according to the book "Intro to probability and statistics" by Rohatgi, the expected value of $X$ is defined as: $$E(X) =...
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2answers
19 views

Two different samples from different time periods.

I have a sample of grades from 1000 students. The average mark was 60 with a standard deviation of 3. A year later I collected a sample of grades from 50 students sitting the same test. The average ...
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2answers
39 views

Probability clarification

Consider the reading habits of the X class. If we know that 30% of the class students read USA Today daily, 40% read Salt Lake Tribune daily and 10% read both of them daily, what is the percentage of ...
0
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1answer
90 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
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1answer
48 views

Geometric sum of geometric random variables

I am looking to find the probability mass function of $Y=\sum_{i=1}^NX_i$ where $X_i\sim\textrm{Geometric}(a)$ and $N\sim\textrm{Geometric}(b)$. I attempted to do this by finding the probability ...
0
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1answer
60 views

Probability Of Birthday Months

In a class of 20 students, what is the probability that at least one will be born in november or december? Using the complement we can look at $P(\overline A)=(\frac{10}{12})^{20}$ so $P(A)=1-(\...
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0answers
19 views

Optimal decision for sampling a distribution.

I was wondering which probability distribution is best sampled with $\pm\alpha^n, n\in\{1,2,\cdots\}$ for various values of alpha. Sampling means to pick the one which is closest, store the sign and ...
1
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1answer
42 views

How to show that $\mathbb{E}(\lim_{n \to \infty} X_n) = 0$ when $X_n(x) := n \cdot 1_{[0,\frac{1}{n}]}(x) \qquad (x \in [0,1])$

from the answer of Exchanging limit and expectation for $L^2$ random variables: Consider for example the probability space $(\Omega,\mathcal{A},\mathbb{P}) := ([0,1],\mathcal{B}([0,1]),\lambda|_{...
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4answers
21k views

A fair 6 sided dice is rolled 4 times. What is the probability that at least 3 of the numbers will be either 1 or 6?

I'd really love a sanity check here as I walk through what I believe is the solution. Total possible outcomes = $6^4 = 1296$ Possible combinations of 3 rolls being either 1 or 6 = $({}_4C_3)\cdot2 = ...
2
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1answer
33 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with $...
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0answers
43 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here (https://en.wikipedia.org/wiki/...
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0answers
27 views

Is the Martingale property still true for $\xi$ not necessarily $C^1$?

Denote $$M(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$ Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ the ...
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0answers
36 views

Copula vs Exprimental Copula

I have read some texts about finding/approximating copulas for a given sample based on known (famous) copulas. My question is: when we have the experimental CDF of (X, Y), why we should try to find a ...
1
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1answer
58 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
1
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1answer
58 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
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1answer
46 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that $\mathcal{A}_1,\ldots,\...
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3answers
118 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
0
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2answers
307 views

Odds of $X$ number of items getting picked twice or three times in a row

There is a list of $63$ items, from which $20$ items are picked in month $1$. What is the chance of $1$ item from the $20$, being picked again in the month $2$?(month $2$ also picks 20 items) Then ...
0
votes
1answer
32 views

Expected value of a sample

I am unsure of how to solve this question. I know from examples questions that expected value of a sample is usually very close to the population mean. However, it says to compute the expected value ...
0
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2answers
440 views

let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?

I have a question about probability: let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is? can someone ...
3
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1answer
131 views

German Tank Problem, Confidence Level

Suppose you're in a city with n cabs. Each cab has a distinctive number from 0 to n. You take a cab 10 times, the choice of the cabs is independent and equiprobable. The cab with the biggest number ...
1
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1answer
36 views

Scale-free property of random graphs

From this Wikipedia page, I gather that when the degree distribution of a graph obeys the power law, the graph is termed 'scale-free'. I would like to know the reason for this term. What has scaling ...
0
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1answer
37 views

identifying sudden change in value given a list of values over time

I have a list of the average price of an item in a game over time. Things don't tend to move much. I am wondering how I can detect whether a new value inserted is a surprising movement in price. I ...
4
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2answers
104 views

No Adjacency Combinatorics Problem via Generating Function

I would like to find the generating function solution for the following combinatorics/probability problem. I have a combinatorial solution and the generating function deduced thereof. But I can not ...
2
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1answer
88 views

Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
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1answer
57 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
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0answers
33 views

This random variable converges in distribution?

$\delta_x$ is a Borel probability that $\delta_x(x)=1$ and $\mu_n$ is a uniform distribution in this interval $(1, 1 + \frac{1}{n})$ The variable $X_n\sim \frac{1}{2}\delta_{\frac{1}{n}} + \frac{1}{2}...
0
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1answer
65 views

Find the probability of opening all the boxes

Suppose there are $20$ boxes which $1-20$ are printed on each box. There is a key in each box which are also marked with $1-20$. So only the key with the same number with the box can open it. For ...
0
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1answer
67 views

Variance of absolute value sums of random variables

Let $X=\left|\sum _{i=1}^n Z_{i} \right|$ and $Y=\sum _{i=1}^n |Z_{i} |$ where random variables $(\textit{Z${}_{i}$})$ are i.i.d, and $Z_{i} =0$,$+1$ or$-1$, with probability ${p}{}_{0}$, ${p}{}_{1}$,...
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2answers
450 views

Puzzle: Guessing the bigger number!

Consider the following interesting puzzle: "Alice writes two distinct real numbers between 0 and 1 on two sheets of paper. Bob selects one of the sheets randomly to inspect it. He then has to ...
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0answers
81 views

Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
1
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1answer
59 views

Borel-Cantelli question

If $X_1...X_n$ are i.i.d. and $\mathcal{N}(0,1)$ how can Borel-Cantelli lemma helps us to proof a.s. of: $$\max\{X_{n^2+1},X_{n^2+2},\dots,X_{n^2+2n}\}\ge5 \text{, }\forall n>N$$ Thank's for your ...
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0answers
72 views

Queue theory - M/D/k - Probability of never having a queue before a time T

This is probably a known result, but I couldn't find any resource pointing directly to the issue I'm trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a ...
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0answers
30 views

construction of a path of quadratic variation

Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= t_{0}&...
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2answers
51 views

How will we find $P(E)$ instead of $P(\bar E)$? [closed]

I want to know this to know why finding probability by finding complement is more easy in this case? Question: A sequence of 9 bits is randomly generated. What is the probability that at least one ...
0
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0answers
67 views

Probability of choosing a point from large set

Let x and y be non-negative integers and $y \le x \le m$. Let us define a function $ f(x) = x/n, n = 1,2,3,...,m $ For a value $ m $, what is the probability of selecting a point $ p(m,y) $ so that $ \...
2
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3answers
45 views

The non-uniform probability of sums from the throw of multiple dice

I'm reading The Drunkards Walk by Leonard Mlodinow. In the book, the author writes: From a throw of three dice, a sum of 9 and 10 can be constructed in an equal combinations. However, the outcome (...
0
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1answer
58 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
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5answers
444 views

Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
1
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3answers
77 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
0
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1answer
29 views

how to find the cumulative density function

Consider $$f(x)=3x^{-4} \qquad \mbox{on} \qquad x\geq 1.$$ Let $X$ be a continuous random variable on $x\geq 1$. Find the cumulative distribution $F(x)$ for $X$. I know that CDF for a continuous ...
0
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2answers
205 views

Tabulate the probability distribution of $x$.

If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? Tabulate the probability distribution of $...
2
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3answers
229 views

If a die is thrown thrice. Find the probability that the largest score is three times the smallest.

I have no idea about the answer, but I'm viewing the question this way; If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be $3$,...