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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What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Ok, so I think I have a working solution to this problem. Heres how I would solve it: so you look at a 6 appearing as a success and everything else as a failure. So from here you can you use the ...
2
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1answer
33 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
2
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1answer
483 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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1answer
23 views

About the equivalence of two asymptotic probabilistic statements

Let $g(n)$ be some monotone increasing function of naturals, and let $X_n$ be a sequence of positive random variables. Consider the following two claims: Claim 1. $\exists f=o(g(n)),\ ...
1
vote
1answer
24 views

p.d.f of the absolute value of a normally distributed variable

I came across this question as an exercise, had a brief idea, but didn't know how to proceed. Let X ~ N(0, 1). What is the p.d.f of |X|? I know the final p.d.f looks just like the right half of the ...
0
votes
1answer
35 views

Proof of infinite monkey theorem.

I was just wondering, does the infinte monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I ...
0
votes
1answer
27 views

Significance level for a hypothesis test for a linear regression

Consider linear regression model $Y_i=a+b\cdot x_i+\epsilon_i$, $i=1,2,3,4,5$, where $a,b\in\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, $\epsilon_i$ are iid, normally distributed with ...
0
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1answer
446 views

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f. for a random variable X. c) f(x) = x/c, x = 1,2,...,n d) f(x) = c/(x+1)(x+2), x = ...
10
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0answers
335 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
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0answers
147 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
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0answers
459 views

Expected value of the distance square

Given two points $X,Y$ on two sides of square $[0,1]\times [0,1]$ ($X:(0,1/2),Y:(1,1/2)$ (PS: My original question is $X,Y$ on opposite of a square, but I think that's not the real case) )and $n$ ...
8
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0answers
136 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 ...
8
votes
0answers
113 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
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0answers
415 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...
8
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0answers
227 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
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0answers
233 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
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0answers
312 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
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0answers
308 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
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0answers
215 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
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0answers
170 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
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0answers
89 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
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0answers
211 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 ...
6
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0answers
602 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 ...
6
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0answers
465 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
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0answers
200 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
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0answers
359 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
301 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
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0answers
45 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) = ...
5
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0answers
89 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 ...
5
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0answers
55 views

Probability of winning at Solitaire

Using a standard deck of playing cards, how many ways of assembling (shuffling) them will result in a competent player always "going out" in a standard (seven initial columns, every remaining third ...
5
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0answers
206 views

Probability that at least one of four hands missing at least one suit

Deal each of four players a 13-card hand at random. What is the probability that at least one of the four hands is missing at least one suit? Let $A_i$ mean that player $i$ is missing at least one ...
5
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0answers
115 views

Probability of transmission between two nodes in a neural network at exactly d timesteps

I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is 1/N, meaning that if node n has fired the ...
5
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0answers
158 views

Coding Theory Problem to save Humanity

For starters, this problem doesn't originate from me, it's a friend's coding theory problem and I got interested, thinking about it, but I can't think of any as I only have very basic coding theory ...
5
votes
0answers
105 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, ...
5
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0answers
205 views

Probability of Multiple Collisions in the Birthday Problem

I need help with an approximation concerning the birthday problem. In a recent MAA Monthly (August-September 2013) article "Simple Approximation Formulas for the Birthday Problem" by Matthias Arnold ...
5
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0answers
278 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
5
votes
0answers
179 views

Central Limit Theorem on the Circle

I am interested in a circular equivalent to the classical CLT. Is there a necessary and sufficient condition telling when a normalized sum of circular distributed random variables converges to a ...
5
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0answers
208 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
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0answers
160 views
+100

A fun card game involving probability, getting all 13 ranks (any suit(s)) vs. 5 in a row of red or black.

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, shuffled well before each hand is drawn, and randomly draw cards from it one a time ...
2
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0answers
42 views

Applying Bayes Rule

I'm currently revising for my exams and something in one of the slides is throwing me off. I don't understand how Bayes Rule was applied on the following: (I understand the first line - thats ...
1
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0answers
22 views

Random variable: $X\sim Normal(m, {\sigma}^2)$, find the characteristic function of $X^2$

Is it possible, knowing that $X$ is a random variable with normal distribution( with parameters $(m, {\sigma}^2)$), to find the characteristic function of $X^2$ ? What I thought is: Since: $\phi(X) ...
1
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0answers
22 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
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0answers
17 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
0
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0answers
12 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
0
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0answers
9 views

Random variables set representation in the sample space

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
0
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0answers
15 views

Serial Number in a Geometric Distribution

I won't bother posting the whole exercise.Basically, we've got 2000 pc's and 12 of them are malfunctioning. At some point, the exercise writes: We choose the pc's until we find a malfunctioning ...
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0answers
30 views

Statistics, measuring unknown mean with standard deviation and probability

So I am stuck on this question and wondering if anyone can give me any hints towards it Estimate the mean pH of rainfalls in an area. Previous studies showed that the standard deviation in the area ...
-1
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0answers
22 views

Why does a number of exponentially distributed variables in time t not follow a Poisson random variable at time t?

Let's say we an explosion happen. Each explosion occurrence is exponentially distributed with parameter lamba. I want to find the number of explosions that happen within time t. Why is this not a ...
-1
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0answers
18 views

Probability of deviation of sample distribution, why absolute difference?

In my textbook is a worked problem as follows: Assume the population mean is 5. Take 100 measurements and we find the sample mean $\overline{x} = 5.027$. Is our assumption correct? In other words, ...
-1
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0answers
25 views

Normal distribution: how to approach a simple problem.

how could I compute this probability? $$P(|X-2|>5)\text{ if }X\stackrel{d}{=}\mathcal{N}(1,4).$$ $|X-2|>5\Rightarrow 7<X<-3\Rightarrow 3<Z<-2$. But calculating the probability to ...