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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Variational distance basic properties

The variational distance between two probability distributions $X$ and $Y$ taking values on the same alphabet $\mathcal A$ is defined as \begin{equation} \delta (X,Y)=1/2\sum_{a\in A} ...
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76 views

Is this a known distribution?

I came across the distribution on $(0,1]$ with the following density function $$f(x) = \frac{2}{\pi}\sqrt{\frac{1}{x}-1}$$ Is this a known distribution? Any references will be appreciated.
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306 views

arbitrary three points on a circle

Three points $A,B,C$ are chosen randomly on a circle.Find the probability that $\angle ABC$ is less than $\theta \in (0,\pi)$. My method of solving this problem is this: Fix a point $B$ on the ...
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223 views

Probability of picked card being a spade or ace

A pack of card is well shuffled and . Top 25 cards are removed. From the remaining cards 14th card from top is picked find the probability of card being an Ace or a Spade.
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282 views

Statistics: How to measure how accurately probabilities are reported?

If you roll a six sided die a bunch of times, and count how many times the number 1 shows up, you'd expect it to show up about 1/6 of the time. Now if you roll this die 1000 times, and the number 1 ...
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296 views

Probability that a short random string, over an N-letter alphabet, appears 'k' times in a longer random string

I have two random strings over an $N$-letter alphabet: one is a shorter $M$-letter string, and one is a longer $L$-letter string. Assuming that two or more instances of the shorter string can ...
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4answers
199 views

Probability of an exact number of duplicate pairs when choosing X from Y.

Here is the problem I'm faced with, as best as I can describe it. There is a set of 256 values (a byte), and 108 values are chosen from this set. Each choice may be any value from 0 to 255. What is ...
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1answer
2k views

Probability a coin comes up heads more often than tails

I am told that a fair coin is flipped $2n$ times and I have to find the probability that it comes up heads more often that it comes up tails. Please, how do I find the required probability?
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373 views

Monte Carlo algorithm that determines if a permutation of the integers 1 through $n$ has already been sorted.

This question is from "Discrete Mathematics and Its Applications", from Kenneth Rosen, 6th Edition. Devise a Monte Carlo algorithm that determines whether a permutation of the integers 1 through ...
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1answer
89 views

how to find the mean of x using chebyshev's inequality?

x, y are independent possion variates. variance of X+Y = 9 P(X = 3/X+Y=6) = 5/54 Can anyone help me find the mean of X? Does Chebyshev's inequality come into picture?
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1answer
879 views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
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2answers
151 views

Given $3$ dice, which number is the most likely to appear

Given $3$ dice, what is the value of the sum of the number of the $3$ dice most likely to appear? I know that by symmetry, there would always be $2$ different values with the same probability to ...
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1answer
51 views

Find $P(b^2\ge4ac)$ given that $a,b,c\in\{-9,-8,\dots,8,9\},a\ne0$

I was doing some review on probability and came across the following exercise: A quadratic equation $ax^2+bx+c=0$ is copied by a typist. However, the numbers standing for a, b and c are blurred ...
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2answers
53 views

Lower bound on a function of probability distribution

Let $$ \gamma = \frac{1}{\sum_{y}f(y)W(y)}, $$ where $$ f(y) = 1 + e^{-|y|} $$ and $W(y)$ is a probability distribution (unknown) with $y \in \mathcal{Y}$ arbitrary but discrete, and $x \in ...
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1answer
113 views

$E[x\mid x>1]$ if $X \sim \exp(\lambda)$

I need to find $E[x\mid x>1]$ if $X \sim \exp(\lambda)$. I first tried: $$f(x|x>1) = \frac{f(x)}{\int_{x=1}^{\infty}f(x) dx}.$$
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1answer
232 views

Expectation of a min function of summation of indicator random variable

Suppose $X_i$ is an indicator random variable. There is another random variable Z defined as $Z = \min(c, \sum_i X_i)$, where $c$ is a constant. How do we compute $E[Z]$? I have come up with the ...
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562 views

Bhattacharyya Coefficient

Let $W(y/x)$ be a conditional probability distribution where $x \in \{0, 1\}$ and $y$ is arbitrary but discrete, then Bhattacharyya coefficient is given by $$ B(W) = \sum_{y} \sqrt{W(y/0)W(y/1)} $$ ...
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376 views

What is the expected area of a polygon whose vertices lie on a circle?

I came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly ...
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1answer
93 views

hash probabilities

Given the 16bit Pearson hash of a 112 bit message, how many other messages have the same hash ? What's the probability that a similar 112b message of a given 112b message (you can define similar ...
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1answer
171 views

Elementary Probability Questions

Toss a coin three times, so event space $\Omega=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$. We win $\$1$ if we flip a Head and lose $\$1$ for a Tail. Let $\mathbb{P}(H) = p$ and $\mathbb{P}(T) = q$. The ...
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1answer
790 views

calculate standard deviation from percentage of mean occuring.

I'm not a math person, although I find it quite interesting. I'm a programmer but I've got a math problem I'm trying to figure out. Lets assume I'm trying to create a program that will predict at ...
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2answers
70 views

Joint distribution of consecutive renewal times

Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. ...
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1answer
681 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
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1answer
186 views

Probability of vertices in a complete bipartite graph being disconnected such that no path of length 2 remains between them?

My problem is the following. I have a set of vertices $N$ and a set of vertices $H$. Each vertex $n \in N$ is connected by means of an edge to each vertex $h \in H$. So the two sets of vertices and ...
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3answers
132 views

Transformation of double-integral with $y-x\leq 1$ and $x-y\leq 1$ for probabilities

Let the the function $f(x,y)$ be given by $$f(x,y)=\begin{cases}cxy,&-1\leq x\leq 0\wedge 0\leq y\leq 1\wedge y-x\leq 1,\\cxy,&0\leq x\leq 1\wedge -1\leq y\leq 0\wedge x-y\leq ...
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1answer
554 views

Distribution of the product of an exponential and a uniform distribution

I'm trying to show that $U(X+Y) = X$ in distribution, where X and Y are independent $\exp(\lambda)$ distributed and $U$ is uniformly distributed on (0,1) independent of $X+Y$. I've been able to show ...
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1answer
120 views

Does this multivariate function have only one maximum?

Let $X_1$ and $X_2$ be random variables (not of the same distribution and not independent). Both have a zero probability of being below $-1$. Their joint density is $\rho(x_1,x_2)$. Also, they both ...
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1answer
146 views

The $\alpha$-Potential-Operator (Definition and resolvent Equation)

during my studies I encountered the following Operator ($X_t$ is the standard Browniang Motion, $\alpha>0$ and $f$ is bounded function ) $U^{\alpha}f(x)=\mathbb{E}^x \int_0^{\infty} e^{-\alpha ...
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1answer
320 views

Combination of a normal r.v. with a log-normal one

It is well-known that a sum of normal r.v.'s is another normal r.v., and a sum of log-normal r.v.'s can be accurately approximated with a log-normal r.v. But what can we say if we have a mixture of ...
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1answer
83 views

Solving this permutation

I know this is an extremely noob question, but I need some help. since I am stuck Prove the formula $$p(n,r) = \frac{(n + 1 -r) \; (r^2 - 3r + 3) \; (r-2)!}{n!}$$ from this answer.
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41 views

Probability question regarding range

I am stuck with the following question. It is as follows, ...
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1answer
359 views

Estimating maximum value of random variable

Suppose I have some random variable $X$ which only takes on values over some finite region of the real line, and I want to estimate the maximum value of this random variable. Obviously one crude ...
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1answer
63 views

Probabilistic regression on outliers

I have a given data set $D = \{ x_i, y_i \}_{i=1}^n$ for a regression problem. When I plot the data, it looks like there is an underlying parabola (2nd order linear model) and some outliers. I want ...
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133 views

Estimate total song ('coupon') number by number of repeats

If shuffle-playing playlist ×100 resulted in [10 13 10 3 2 2] different songs being repeated [1 2 3 4 5 6] times, what is the estimate for the total number of songs? (assuming shuffle play ...
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2answers
728 views

Flipping Cards Probability

You have a deck of cards, 26 red, 26 black. These are turned over, and at any point you may stop and exclaim "The next card is red.". If the next card is red you win £10. What's the ...
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4answers
497 views

Solving for $E(X^2)$ if I know $E(X)$

I am trying to find the variance but I don't know how to calculate $E(X^2)$, but I do have a process that will enable me to find $E(X)$. How can I find $E(X^2)$? In my case I have two 6-sided dice, ...
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1answer
153 views

What is the probability of two people sitting side by side each catching a home run in the same game?

Yesterday, in Edmonton, at a baseball game two of my friends caught a home run, in separate plays during the game, and I am wondering where to begin in analyzing what the probability of this happening ...
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0answers
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Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
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1answer
1k views

What is the probability of two people meeting?

I am trying to figure out a solution to the following problem: Let there be two groups of people, Group A and Group B. Group A represents x percent (e.g. 1%) of the world's population, and Group B ...
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1answer
1k views

How to derive Marginalization? [closed]

How would you derive marginalization as it is given here: http://en.wikipedia.org/wiki/Marginal_distribution? Thanks,
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139 views

Random variables and sigma field

Given $Z_1,Z_2,\cdots$ i.i.d with $E|Z_i|<\infty$. $\theta$ is an independent r.v. with finite mean and $Y_i=Z_i+\theta$. If we define $F_n=\sigma(Y_1,\cdots,Y_n), F_\infty=\sigma(\cup_n F_n).$ ...
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2answers
344 views

Expectation Values inside absolute value operator

first: are these equality true ? $$|E[Y]-E[X]|=|E[Y]|-|E[X]|.$$ $$|E[Y]-E[X]|^2=|E[Y]|^2-|E[X]|^2$$ second: what is result of this relation: $$\sum_{i=1}^{3}p_i.(X_i-\mu)^2=?$$ where the $\mu ...
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122 views

Help with probability question.

If ten people each have a ten percent chance of winning a prize. What is the probability that at least one of them wins the prize? Background : there are 100 prizes to be won, and 1000 people with ...
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392 views

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let ...
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2answers
134 views

A wrong reasoning about conditional probability

Two of three prisoner A, B and C will be executed, A asks the name of one other than A himself who will be executed. Jailer says that it is B. Merely by asking the question, A reduced the probability ...
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1answer
184 views

Basic calculation of an expectation

We define $J_{k,n}:=((k-1)2^{-n},k2^{-n}]$ for $n\in \mathbb{N}_0$ and $k=1,\dots,2^n$. Let $W$ be a Brownian Motion. Let $n\ge m$ and we assume $J_{k,n}\subset J_{l,m}$. W.l.o.g $J_{k,n}$ lies in the ...
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156 views

Characterising argmax of uniform distributions

I was thinking about comparisons of uniform random variables of the type $U(0,T)$, when I began to wonder about the argmax. Consider a sequence of parameters $T_1\le T_2\le\ldots T_n$ and ...
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659 views

Basic probability problem

Problem states: Consider two events $A$ and $B$, with $P(A) = 0.4$ and $Pr(B) = 0.7$. Determine the maximum and the minimum possible values for $P(A \& B)$ and the conditions under which each ...
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432 views

A question regarding drawing balls of differing colors from an urn before a certain number of draws occur without replacement.

Suppose that the contents of an urn are $w$ red balls, $x$ yellow balls, $y$ green balls, and $z$ blue balls collectively, where $w \geq 3$, $x\geq 1$, $y\geq 1$, and $z\geq 1$. We draw balls randomly ...
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1answer
128 views

Why is $X_1 + X_2 +\ldots + X_n$ a martingale?

If we have $X_k$ random variables with average $0$ and independent, why is the $\sum_{k=1}^n X_k$ a martingale for the sigma algebra $\mathcal F_n$ generated by $\{X_1,\ldots, X_n\}$? I basically ...