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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4
votes
1answer
42 views

shifted exponential distribution with inter-arrival time

Given that time interval $T^*$ in seconds between certain events has a negative exponential distribution. The instrument cannot detect intervals which are less than $\delta$ seconds. Let $T_1, ..., ...
2
votes
1answer
30 views

Expected value of room enters

I was looking at previous exam questions, but one of the questions I don't know how to solve correctly. In this question I need to calculate the expected amount of rooms the mouse enters before he is ...
2
votes
1answer
35 views

Finding an isotropic joint density from a marginal

I'm trying to find out weither it is possible or not to recover an isotropic bivariate pdf from one of its marginal pdf. By isotropic, I mean that the density only depends on the radius when ...
2
votes
1answer
336 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
2
votes
1answer
33 views

Random Variable Vs. Probability Function Intuition?

In a book on probability I'm reading they begin by defining random experiment, outcome, sample space & event, then using these notions they define & a probability space in terms of the sample ...
0
votes
1answer
12 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 $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
0
votes
1answer
23 views

independence of random variable

Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables. 1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are ...
0
votes
1answer
22 views

Complement of Conditional Probability

I'm currently reading this paper Censored Exploration and the Dark Pool Problem and have difficulties in understanding the following simple equality: Let $S$ be a positive integer random variable. ...
0
votes
1answer
350 views

I am trying to solve probability that 2 events do not occur

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the ...
11
votes
0answers
225 views

Probability of two zero inner products

Consider a random $n+1$-dimensional vector $v$. Each $v_1,\dots,v_n$ equals $1$ with probability $1/2$ and $-1$ with probability $1/2$ independently. We also set $v_{n+1} = v_1$. Now consider a ...
10
votes
0answers
218 views

Shooting bullets

This is from http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/May2014.html Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1. The ...
10
votes
0answers
354 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 1

I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ...
9
votes
0answers
153 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
9
votes
0answers
232 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
8
votes
0answers
117 views

Algorithm to compute fastest method of collecting $k$ re-spawning items which spawn at $n$ specified points

Let $V = v_1, \dots, v_n$ be the locations the items can spawn at, and let $U = u_1, \dots, u_k$ be the current positions of the items. We will assume a new items spawns instantly every time we ...
8
votes
0answers
236 views

Extracting an (almost) independent large subset from a pairwise independent set of Bernoulli variables

Let $n>1$, and let $X_1,X_2, \ldots ,X_n$ be non-constant random variables with values in $\lbrace 0,1 \rbrace$. Let us say that a subset of variables $X_{i_1},X_{i_2}, \ldots,X_{i_d}$ is complete ...
7
votes
0answers
250 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
7
votes
0answers
655 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
7
votes
0answers
322 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
7
votes
0answers
309 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
6
votes
0answers
96 views

Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
6
votes
0answers
64 views

Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \} $ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = ...
6
votes
0answers
246 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
6
votes
0answers
210 views

Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)

I have the following problem from book "Introduction to Probability", p.32 A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. ...
6
votes
0answers
95 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
6
votes
0answers
111 views

Random point distribtion

How to generate numerically a set of random points $(x_1,y_1), (x_2,y_2),\cdots, (x_N,y_N)$ such that the pair-wise distances $d = \sqrt { (x_i-x_j)^2 + (y_i-y_j)^2}$, for all $ 0<i\le N, ...
6
votes
0answers
497 views

How does the answer to Feynman's Restaurant Problem change if $M$ is not restricted to a single value?

First, the background: Feynman's restaurant problem asks how we can maximise the total rating of the meals we eat at a restaurant with $N$ items on the menu, given that we know up-front that we are ...
6
votes
0answers
210 views

Question on Conditional expectation

Let $X_1$ and $X_2$ be two random variables on $(\Omega,\mathcal{B},P)$. Suppose there is a function $g:\mathcal{B}\times\mathbb{R}\rightarrow[0,1]$ such that for any $x$, $g(\cdot,x)$ is a ...
6
votes
0answers
364 views

Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
6
votes
0answers
311 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
5
votes
0answers
35 views

Equivalent definitions of Poisson process

Define a Poisson process with parameter $\lambda$ is a counting process $(N(t))_{t\ge 0}$ such that: (i) $N(0)=0$; (ii) It has independent increment property; (iii) $N(t+h)-N(t)$ has Poisson ...
5
votes
0answers
58 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
5
votes
0answers
103 views

Probability that a five is seen before any of the even numbers are seen

A fair die is repeatedly tossed. What is the probability that a five is seen before any of the even numbers are seen? I have my own solution below and just want someone to verify it. According ...
2
votes
0answers
24 views

Stable distributions and equivalence of certain definitions

There are several definitions of stable distributions. The most ubiquitous is arguably that if $X, X_1, X_2, \ldots $ are i.i.d. random variables with probability distribution $F $ then, $F $ is ...
1
vote
0answers
11 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= ...
1
vote
0answers
30 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{\frac{5}{9}}$ in distribution, where $Z_n ...
1
vote
0answers
19 views

Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
0
votes
0answers
6 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 ...
0
votes
0answers
19 views

Distribution of the minimum of two exponential random variables

$X$ and $Y$ are two exponential random variables with rate 1 and 2. lets define random variable $Z$ such that: $z_i = min(x_i,y_i)$, where $i =1,2,3,...N$. Let $V$ be another random variable and ...
0
votes
0answers
22 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 ...
0
votes
0answers
54 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 $ ...
0
votes
0answers
22 views

Find/estimate variance

Let $w_{11},\ldots , w_{nm}\in [0, 1]$ be a set of constants and $H_1(t), \ldots , H_m(t)$ be some cumulative distribution functions (CDFs). Consider a sample of independent random variables $\xi _1, ...
0
votes
0answers
22 views

Calculating expectation of random vector

Let $\Omega=\Theta\times \Pi$ be a finite sample space and $P$ be prob. measure on $2^\Omega$. We define random vector $X: \Omega \rightarrow \mathbb{R}^n.$ How can I calculate following conditional ...
0
votes
0answers
28 views

Notation in probability theory: conditional on multiple events or joint of event with an conditional one

It might be a quite dumb question and if so, I apologize in advance (I am kind of a newbie in probability theory ). But once in a while it bothers me and I can't find the answer to it. Ok, now the ...
0
votes
0answers
18 views

Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
0
votes
0answers
18 views

Bayesian Updating - plug in previous posterior for prior?

Let's say I have two sequences of observations, $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$. For each sequence I'm going to estimate the probabilities of certain events occurring, namely event $A$ in ...
0
votes
0answers
28 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
0
votes
0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
0
votes
0answers
34 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} & & ...
0
votes
0answers
503 views

Conditional Expectation for Geometric Series - Dice problem

A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to obtain a 5 and $Y$ the number of rolls needed to obtain a 6. Calculate $E[X \mid Y=2]$. I found a similar post of this ...