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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7
votes
2answers
317 views

Expected length of a sequence that contains all words of a given length.

Fix some alphabet $\Sigma$ and a positive integer $n$. What is the expected number of random letters drawn from $\Sigma$ until all length-$n$ words are present? For example, let $\Sigma = \{0,1\}.$ ...
0
votes
1answer
141 views

Moment generating functions and normal random variables.

Suppose that the distribution of students' grades in a probability test is normal, with mean 72 and variance 25. i.) What is the probability that the average grade of a class with 25 students is 75 or ...
2
votes
1answer
1k views

Confusion about Banach Matchbox problem

While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand. The problem statement is presented below (Source:Here) ...
2
votes
1answer
60 views

Show that $\sigma_{X+Y} \leq \sigma_X + \sigma_Y$

Show that $\sigma_{X+Y} \leq \sigma_X + \sigma_Y$ I tried using the formula $Var(X+Y) = Var(X) + Var(Y) + 2\sigma_{XY}$ and also the fact that the correlation coefficient, defined as ...
2
votes
1answer
65 views

Proof of $E(E(X|Y))$

Why is $$ E(E(X|Y)) = \int_{\infty}^{\infty} E(X|Y)f_2(y)dy $$ So, another way of putting it, is why is this, given $$ E(E(X|Y)) = E(h(y)) $$ Why do they take the above function to be true? And a ...
2
votes
1answer
46 views

Picking from a set probability question

If I had a set with $x$ elements and I read a random one $y$ times (picking with replacement), what would the average number of unique elements I had read as a function of $x$ and $y$? This is very ...
0
votes
1answer
297 views

Calculating coefficient of generating function with Coin

The problem I'm currently looking over requires use of generating functions to solve the following: If a coin is flipped $25$ times with eight tails occurring, what is the probability that no run of ...
4
votes
2answers
187 views

3 Die Probability

The number of times you have to roll a set of 3 die to see a 6 on each die appears to be around ~10.555 according to a implementation of this in c++ and c#. How would this be statistically proven as ...
0
votes
1answer
38 views

Question on $\lim\limits_{n\to\infty} P(|Y_n| \geq c) = \lim\limits_{n\to\infty} \frac 1n$

Consider a sequence of discrete random variables $Y_n$ with the following distribution: $$P(Y_n = y) = \begin{cases} 1 - \frac 1n, & \text{for } y = 0, \\ \frac 1n, & \text{for } y = n^2, \\ ...
1
vote
2answers
84 views

Is it true that $\lim_{n\to\infty}E[X_n] = 0$ if $X_n\to 0$ in probability?

Is there any counter example that: Let $X_1, X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $c > 0$ $$\lim_{n\to\infty} P(|X_n - 0| > c) = ...
2
votes
1answer
45 views

Variance inequality to show deviation from midpoint

How to show this inequality: If $\mathbb P (X \in [a,b]) = 1$, then $\operatorname{Var}(X) \leqslant \frac{(a-b)^2}{4}$. Thank you!
1
vote
1answer
98 views

A word problem concerning probability. Its pretty interesting

I found an interesting word problem. Check out this link: http://www.math.hmc.edu/~ajb/PCMI/pcmi12.pdf It is problem $A7$. I have been working on it and have reached the following conclusion: Say ...
1
vote
0answers
42 views

identification of conditional joint density from conditional marginal densities

I have the following factor model: $W_1 = \theta + \eta + \nu_1$ $W_2 = \mu_2(\theta)+\eta+\nu_2$ where $\eta$, $\nu_1$, and $\nu_3$ are mutually independent given $\theta$. I already know the ...
3
votes
1answer
89 views

Basic Question about linearity of expectation

I am going through some introductory notes on probability here http://www.stat.berkeley.edu/~aldous/134/gravner.pdf In Chapter 8, page 89, there is a problem where you get a bag containing 10 Black, ...
0
votes
2answers
116 views

HELP Distribution of the Minimum of two random variables

Well Let $Y$ be a random variable that could be discrete or continuous and $M$ a positive constant random variable Find the distribution of $S$$=$$min${Y,M} My progress so far is : $p($S ...
0
votes
1answer
435 views

Rat in Maze Probability

I am trying understand what am I missing in my way of solving rat in maze problem... The question and solution is given in this link http://www.ams.sunysb.edu/~jsbm/courses/311/rat-in-maze.pdf ...
1
vote
2answers
1k views

$P(Y<.5 | X >.5), P(Y>2X), P(.5<X+Y<1.5)$ with Joint probability density function.

Let X and Y have the joint probability density function $f(x,y) = \frac{3}{2}(x^2 +y^2)$, $0<x<1,0<y<1$ a.) Find $P(Y<.5|X>.5)$. My answer is $\frac{5}{11}$. b.) Find ...
1
vote
1answer
102 views

Time Periodic Homogeneous Markov Chain

I want to find a textbook or survey article reference with a treatment of discrete-time, inhomogeneous, yet time periodic, markov chains on finite state spaces. Elaboration: I have an inhomogeneous ...
1
vote
3answers
94 views

Apparent contradiction for probability density functions?

Consider a probability density function $\it{pdf}$, $f\left(x\right)$, which can be expanded as: $$ f\left(x\right) = \sum_{k=1}^{\infty} \alpha_k \delta\left(x-x_k\right)$$ It is easy to verify, by ...
1
vote
4answers
165 views

Birthday paradox: meaning of random

In the wikipedia page (http://en.wikipedia.org/wiki/Birthday_problem) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some ...
0
votes
1answer
67 views

Lower Expectation

Let $X$ be, for simplicity, a finite set (with the discrete topology). Denote with $M(X)$ the set of probability measures on $X$ endowed with the weak topology. For $\mu\in M(X)$ and a (necessarily ...
0
votes
1answer
300 views

Approximating Chi squared distribution

A machine in a heavy equipment-factory produces steel rods of length Y , where Y is a normally distributed random variable with mean 6cm inches and variance $\frac{1}{4} cm^2 $. Thecost C of repairing ...
0
votes
2answers
84 views

Summation problem in an expected value question

A fair die is successfully rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find E(X|Y=5). My attempt is: E(X|Y=5) = 1*P(X=1|Y=5) + 2*P(X=2|Y=5) + ...
0
votes
1answer
54 views

Need help to create formula/equation

I am looking to try create a formula/equation(I am a novice). I'll use a fictitious example that has to do with basketball. Assume there are 5 players that score in each game for each team. Team A: ...
0
votes
1answer
224 views

Motivation for Measure Theory example

I was taking a look at this book while trying to pick a book for learning some rigorous probability theory. I have been totally stumped by the motivating eg. on the first page. Specifically, I am ...
1
vote
1answer
124 views

probability: random variable

From the Ross book ex.13 chapter 4: A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to sale with probability $0.3$ and the second will lead ...
0
votes
2answers
88 views

Inner product in the Hilbert space

For the $L^2$ space we define inner product as $\langle X, Y\rangle = E[XY]$. With respect to which density this expectation is taken, $E[XY] = \int xyf(x,y)dxdy$ or with respect to the marginal ...
0
votes
1answer
24 views

look for a metric for a two variable system

I have a series of experiments for different objects from the experiments let me put it in an abstract way there is a condition A for a specific object O the success rate/percentage is p(A) generally ...
3
votes
0answers
84 views

Quick way to tell if a set of dice is NOT non-transitive

Is there a quick way to tell if a set of six-sided dice cannot be non-transitive? I've writing an algo and brute force is taking too long to find out. I had a look at ...
0
votes
1answer
156 views

How many samples do I need to estimate the click-through rate confidently?

In online advertisement, Click-Through-Rate = Clicks / Impressions How many clicks or impressions do I need that I can confidently say our tomorrow's CTR is ...
2
votes
2answers
300 views

Probability of an event occuring n times before its complement occurs m times

Let the probability of event $E$ be $p$, and let $F$ be the complement of event $E$, so probability of event $F$ occurring is $1-p$. What is the probability of event $E$ occurring $n$ times before ...
1
vote
0answers
25 views

What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]

Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads? How do I apply Chernoff Bounds to this? I really need help understanding Chernoff Bounds.
0
votes
1answer
176 views

Question regarding Type II Error in Hypothesis Testing

The following is a homework problem and I am not really sure where to begin or how find what the question is asking. Suppose that one observation from the exponential pdf $f_{y}(y)=\lambda ...
1
vote
1answer
138 views

Help evaluating long definite integral.

First, thanks for your time and input. In some research, I evaluate the following integral $$\int_0^1 x (n-1) n v^{-2n}(v-x)\lambda^{-n}(v-2x+\lambda v)(\lambda v-x)(x(v+\lambda v-x))^{n-2} dx$$ ...
2
votes
2answers
528 views

The Carnival Dice game

I was reading through an elementary mathematics book lately. The author claims that every probability question, no matter how intimidating it may look, can be solved using a four-step method: ...
2
votes
2answers
96 views

picking 10 different numbers from 1-50

Stage 1: picking randomly 10 different numbers from 1-50 and writing them down. Stage 2: Then, randomly, picking 10 other different numbers from 1-50. What is the probability that the sum of the ...
4
votes
2answers
1k views

Rolling a fair die 4 times, what is the probability of getting an increasing sequence of numbers?

Game: I roll a die 4 times. What is the probability that I get a strictly increasing sequence of numbers. My initial thought is as follows: we condition on R1 (the first roll) being a 1, 2, or 3 ...
2
votes
2answers
107 views

Probability of line intersecting the convex set.

I would like to prove this theorem: Let $A,B \subseteq \mathbb{R} ^3$ be convex, limited sets. $B \subseteq A$. I have a "random line", which intersects A. Probability, that this line also intersects ...
1
vote
1answer
116 views

Find $E[X^2 + Y^2]$ for the given joint density function.

Let X and Y have joint probability density function $f(x,y) = \frac{3}{2}(x^2 +y^2)$, with $0<x<1, 0<y<1$. Find $E[X^2+Y^2]$. I came up with $\frac{14}{15}$ by integrating ...
1
vote
1answer
47 views

A flea jumps on a regular $n$-gon

A flea jumps on a regular $n$-gon with center $O$ with the following rules: it starts in $O$; if it's in $O$ it can jump to any vertex (with equal probability); if it's on a vertex it can jump in $O$ ...
4
votes
1answer
542 views

Proof of Khintchine's inequality

I'm trying to understand the proof of Khintchine's inequality in these lecture notes: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf On page 27, second display-style equation after (51), the ...
2
votes
1answer
103 views

Using empirical density function as an estimator of a given probability density

We know empirical distribution function is defined as $F_n(x)=\frac{1}{n}\sum\limits_{i=1}^nI(X_i \leq x)$. Then define empirical density function as $ f_n(x) = \frac{F_n(x+b_n)-F_n(x-b_n)}{2b_n} $ . ...
1
vote
1answer
203 views

Expected value of a Poisson sum of confluent hypergeometric functions

How to compute the expected value of a Poisson sum of the following confluent hypergeometric function: $$ \sum_{y=1}^{Y} {}_1F_1(y,1,z) $$ where y is positive integer taking values from the Poisson ...
-1
votes
1answer
58 views

Positive Outcome!

I have a question on Probability. With two dices, which each have six sides people are making duels with these dices. the winner is the one who rolls the highest. the chances are of a 50% win as it is ...
-1
votes
2answers
47 views

Problem about convergence in Probability (2) [duplicate]

Let $X_1,X_2,\dots$ be a sequence of random variables with $$ \lim_{n\rightarrow+\infty}E\left[\left|X_n\right|\right]=0 $$ Is it true or false that the sequence $X_n$ must converge to $0$ in ...
0
votes
1answer
148 views

Regarding properties of non-transitive dice

First I'm not a mathematician so please ask for clarifications. Is it required condition for the set of non-transitive dice to have numbers that are not common amongst themselves? So for example ...
1
vote
1answer
102 views

How to sample uniformly from an $\epsilon$ ball?

Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed ...
0
votes
1answer
104 views

The hiring problem extension

Some Background: Some instance of the Hiring problem is as following: let $a_1$,...,$a_n$ be candidates for a job in some company, assume the company interviews one of them at a time, and their ...
2
votes
1answer
253 views

Can one sample uniformly from the surface of an $n$-sphere of non-unit radius using normal r.v.'s?

One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables ...
1
vote
0answers
60 views

Probability function (Very simple indoor positioning system)

I am a programmer, I am working on a masters thesis, tts an Android application, where I will need to implement a very simple indoor positioning system using WiFi RSS. The number of routers is fixed. ...