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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134 views

Example use of Chebychev’s inequality

Question: A number in the interval $[0, 4]$ is selected randomly. How many picks do you have to make so that the arithmetic mean $X$ satisfies $P[|X-2|\ge 0.1]\lt 0.01$ ? Answer: I've solved by using ...
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1answer
142 views

Lower bound expected value of n-th root

I'm faced with the following problem: I have to lower bound the expected value of the n-th root of an arbitrary distributed real random variable using its expected value. So I'm looking for something ...
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2answers
195 views

Are the first flip and second flip of this problem independent events?

Problem Given 5 coins: 2 double-headed coins 2 fair coins 1 double-tailed coin A coin is chosen at random and flipped. The coin is then flipped in a second time. ...
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3answers
526 views

Am I over thinking this 5th grade coin flipping homework problem?

Looking over my daughter's homework, I had a question that seems best to be asked here. It's from Everyday Mathematics (Volume 1, Grade 5, p 42.) Tim flipped a coin 10 times. It landed heads up ...
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3answers
332 views

What is “sampling from a distribution”?

Exercise 4.11.3 of Grimmett and Stirzaker's Probability and Random Processes reads "Use the rejection method to sample from the gamma density $\Gamma(\lambda,t)$ where $t (\geq 1)$ may not be assumed ...
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1answer
120 views

How is this form of the Chernoff bound derived from the other?

Looking over the Wikipedia page for the Chernoff bound, it's given at the top as $P \geq 1-\mathrm{e}^{- 2n \left( p - \frac{1}{2} \right)^2}$, where $P$ is the probability that a majority of biased ...
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1answer
354 views

Probability of a binary event

Let's say we have a parameter $r$ and a binary event $A$ repeatedly happens. The event is binary, so the outcome is either $0$ or $1$. We have collected a lot of data of the form ...
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2answers
180 views

Probability of independent events

If the probabilities that three children X, Y ,Z will get a ticket for a football game are 0.4, 0.3, 0.2 respectively, calculate the probability that, (Assume that the events of X,Y,Z are ...
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1answer
103 views

With what probability is this polynomial equal to zero (mod a prime $p$)?

If we suppose that we have a polynomial $q(x)$ of the following form: $q(x) = \sum_{i=0}^N{c_i x^i} \text{ where } c_i=0 \text{ or } c_i=1$ In other words, if we are given a polynomial with binary ...
2
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1answer
274 views

Conditional Entropy is less than entropy

Let $p_{ij}=P(X=i,Y=j)$ be the joint distribution, $P(X=i)=p_i=\sum_j p_{ij}, P(Y=j)=q_j=\sum_i p_{ij}$ be the marginal distributions, and $p_{i|j}=\frac{p_{ij}}{q_j}$ be the conditional distribution. ...
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2answers
95 views

Calculating $E[N^2]$ using conditional expectation

Considering the standard geometric coin toss, let $N$ be the number of tosses until the first heads appears. The probability of getting heads is $p$. I understand how to find $E[N]$ by conditioning on ...
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1answer
175 views

Probability of getting $4$ smallest Balls in $2$ Boxes

I have $2n$ balls labeled $1, 2, \ldots, 2n$, and two boxes, Box $1$ and Box $2$. I take $n$ of the balls at random without replacement and place them in the Box $1$. I take the remaining $n$ balls ...
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3answers
116 views

Understanding conditional probabilities in Bayes classifiers in the wikipedia page example

I'm using the wikipedia page example: Why is $P(\text{height} \mid \text{male}) = 1.5789$? This means the probability of height given male? The talk page has a similar question, unanswered. the ...
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1answer
255 views

Show that $P(X+Y+Z\text{ is a multiple of }3)\ge 1/4$

Suppose a box contains tickets, each labeled by an integer. Let $X,Y$ and $Z$ be the results of draws at random with replacement from thw box. Show that no matter what the distribution of numbers in ...
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2answers
525 views

A coin is flipped when dice hits 6 (conditional probability)

Prof gave us homework on conditional probability that is due on the day of the lecture on conditional probability. Yeah, this has been a bad week and I've no idea what I'm doing. Q: 3 dice are ...
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2answers
74 views

Bounds on expected value and distribution of a product of beta random variables

Let $V_1,...,V_n$ be random variables distribution according to the Beta distribution with parameters $\mathrm{Beta}(1,\alpha)$. Define $X_i = V_i \prod_{j=1}^{i-1} (1-V_j)$ for $i=1,...,n$. Is ...
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3answers
727 views

Baby Shower Problem. Too hard for 1st grader but got parents thinking

So our six year old son comes home from 1st grade with the following math puzzle. Your Aunt is having a baby. You have created a party game for a baby shower. It is called pick the gender. You ...
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1answer
411 views

Relative entropy is non-negative

Let $p=(p_1,\dotsc,p_r), q=(q_1,\dotsc,q_r)$ be two different probability distributions. Define the relative entropy $$h(p||q) = \sum_{i=1}^r p_i (\ln p_i - \ln q_i)$$ Show $h(p||q)\geq 0$. I'm given ...
2
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1answer
99 views

Branching process: show $P(Z_m>k|Z_n=0) \le (G_n(0))^k, m<n,k\ge 0$

Let $(Z_n)_{n \ge 0}$ be a branching process with generating function $G_n(s)=E(s^{Z_n})$. Let $m<n$, show that $P(Z_m>k|Z_n=0) \le (G_n(0))^k, k \ge 0$. I know that the event $\{Z_n=0\}_{n \ge ...
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1answer
67 views

Variation over univariate Schwartz–Zippel lemma

Let $n\in\mathbb{N}$ and let $q\in[n,2n]$ be a prime number. In addition, let $s,s':\mathbb{F}_q\to\mathbb{F}_q$ be polynomials of degree $\sqrt{n}$ such that $s\neq s'$. From the ...
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2answers
518 views

Probability Range?

How do you find the range ( min and max) for a probability function such as $$\frac{P (B|A) − P (B)} {1−P (B)}\;?$$ What I tried was to use Venn diagrams, but I couldn't find a solution as the ...
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4answers
480 views

Variance over IID a random number of times

Let $X_1, X_2,\dotsc$ be independent and identically distributed with mean $E[X]$ and variance $VAR[X]$. Let $N$ be a non-negative integer-valued random variable independent of the $X_i$'s. Show $$ ...
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2answers
446 views

Probability of collecting all 4 different items while picking 1 random item from the set

a.) There are $4$ distinct items in the set. What is the probability of picking all $4$ items after picking $n\ge4$. b.) How many items do you need to pick to collect all four with a probability of at ...
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4answers
7k views

Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
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1answer
79 views

Given enough DNA samples, would it be possible to reconstruct the entire genealogy tree of humanity? [closed]

DNA samples of live individuals. It would be more of a mesh/graph than a tree but you get what I mean. I guess we could only have access to potential graphs with varying degrees of probability. In ...
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2answers
508 views

Expected number of rolls until a number appears $k$ times

Let $N$ be the number of rolls until the same number appears $k$ consecutive times. Show the expected value $E[N]=\dfrac{6^k-1}{5}$. I've tried conditioning this on the first occurrence of the ...
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2answers
2k views

Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT

Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
0
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1answer
43 views

The proportion between distinct labels in a multiset and the total amount of labels

Say we have a (multi)set $\alpha$ of $n$ balls, each of them is labeled with a number in $\{1,\ldots,m\}$ (where $m<n$ ). Denote by $d$ the amount of distinct labels in $\alpha$. Is it true that ...
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2answers
362 views

Discrete Math - Calculating the number of different functions possible

Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$. Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$. a.) What is the probability that a random function will be a bijection when ...
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3answers
469 views

Question regarding Conditional Probability with a deck of card

Problem A deck of card is shuffled then divided into two halves of 26 cards each. A card is drawn from one of the halves, it turns out to be an ace. The ace is then placed in the second ...
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1answer
70 views

Is it possible to get the coefficients of the power series

If we have a function $f(s)$ with this form: $$ f(s) = \sum_{i=0}^{\infty} p_i s^i $$ We also know that: $$ f(1) = 1 $$ and $$ p_i \ge 0 \quad \text {for all $ i \ge 0$} $$ Assume we can ...
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2answers
448 views

Baseball betting and probablity

Here is a question that came up during class discussions on Friday: Your favorite baseball team is playing against your uncle's favorite team in the World Series. At the beginning of each game, you ...
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3answers
403 views

What are the odds of rolling a 3 number straight throwing 6d6

If you throw six fair dice, what are the odds that at least three dice make a straight (i.e. 123, 234, 345, or 456) I am certain that I am making a mistake in calculating it?
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0answers
109 views

optimize the expected value of a process

There are $a_i$ balls painted with number $i$. For example if we have balls painted with 1,1,1,3,2. we have $a_1 = 3$, $a_2=1$, $a_3=1$. In total there are $m$ balls painted with number $1,\ldots,n$. ...
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2answers
119 views

Absolute error loss for a gamma random variable

Let $X \sim \operatorname{Gamma}(2,1)$, I would like to minimize with respect to $a$ $$E|aX-1|=\int_0^{1/a}(1-ax)xe^{-x}dx+\int_{1/a}^\infty (ax-1)xe^{-x}dx$$ Is there some neat way to do this? The ...
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2answers
230 views

MSE estimator of MMSE, E[Y|X]

I have the following joint pdf $f(x,y)=2\exp(-x-y)$ , for $0<x<y< \infty$. I found these quantaties: $f(x)=2\exp(-2x)$ $E[Y|X]=x+1$ $E[X]=1/2$ $E[Y]=3/2$ $E[XY]=1$ $E[Y^2]=7/2$ MSE ...
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2answers
378 views

Doubling Money Game

The casino offers a certain win-lose game, where you have $p$ chance of winning. You can bet any amount of money, and if you win you get twice your bet; otherwise, you lose your bet. If you use the ...
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0answers
130 views

Two identities in probability

I ma reading the book An introduction to stochastic processes in physics by Don Stephen Lemons. I have a question on two identities. One identity is the identity (B.3) in Appendix b page 102. How ...
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4answers
549 views

Distribution of maximum of partial sums of independent random variables

I have been looking all over the net to find a way to work out a probability distribution of a maximum of partial sums of independent random variables, but to no avail. So I have decided to try to ...
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3answers
2k views

Probability question about married couples

If four married couples are arranged in a row, what is the probability that no husband sits next to his wife? Would it be $1- \frac{2(4!)}{8!}$?
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2answers
279 views

P(X + Y < a | Y < b) where X and Y are independent continuous random variables

I’m looking for an online source/article/lecture notes/ text book that would contain a detailed/rigorous discussion/explanation/proof of the following result, which was used in Conditional normal ...
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1answer
433 views

Ordering of a deck of cards

If you shuffle n cards as follows: Go through the deck one card at a time and at each card, flip a fair coin. If the coin comes up head, then leave the card where it is, and if it comes up tails move ...
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1answer
95 views

Expectation value of products

I am not sure what the expression of E[XY] looks like given that X and Y are random variables on a finite probability space. That's all I need help on. Thanks!
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1answer
410 views

A Inequality between Bhattacharyya distance and KL divergence

I was trying to solve the following problem. But I dont know how to proceed. I would be really grateful if anybody would point me in the right direction. Let $P = (p_1,p_2, \cdots, p_n)$ be a ...
5
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1answer
234 views

Asymptotic behavior for the return to zero of a simple random walk

I got stuck today trying to understand an argument of the Frank den Hollander Book's. The problem is described below. Let $S_n=\sum_{i=1}^n X_i$ be the simple random walk in $\mathbb{Z}^d$, that is ...
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0answers
57 views

Regularity property of tournaments

Let $T$ be a tournament on $X_n=\lbrace 1,2, \ldots ,n \rbrace$ (i.e. an asymmetric relation $<_T$ on $X_n$). Erdos's famous $S_k$ property says that for any $k$ elements $i_1<i_2< \ldots ...
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2answers
370 views

Expectation of supremum

Let $x(t)$ a real valued stochastic process and $T>0$ a constant. Is it true that: $\mathbb{E}[\displaystyle{\sup_{t\in [0,T]}} |x(t)|] \leq T \displaystyle{\sup_{t\in [0,T]}}\mathbb{E}[|x(t)|]$ ? ...
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2answers
411 views

Joint pdf of X and Y with absolute value range

I have the following joint pdf: $f(x,y)=0.5$ where $0 \leq|x|\leq|y|$, $0 \leq|y|\leq1$, and $0$ otherwise The question is: are $X$ and $Y$ independent and uncorrelated? I know that if ...
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2answers
1k views

What is the probability of rolling 2 7's before 6 even numbers on successive rolls of a pair of fair dice?

Would this equal Pr( rolling 2 7's)/Pr(rolling 2 7's or rolling 6 even numbers)? Or could I approach the problem as follows: 1- (Pr(rolling 6 even numbers)+ Pr( rolling 5 even numbers, 7 and an even ...
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2answers
128 views

Dependence of certain random variables

Consider $X_1,X_2$ i.i.d. standard normal random variables(mean 0, variance 1). Are the random variables $Y=X_1+X_2$ and $Z=X_1-X_2$ dependent? I am not sure how to prove this one way or the other.