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Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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843 views

Economics Practice Homework Problem

Working on probability theory in an economics class. This was on of the practice problems that I'm struggling to back up with math. Any help would be appreciated! :) An accident insurance company has ...
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3answers
509 views

Expected time of tree search algorithm on random input

We have a perfect binary tree on 2^k-1 nodes. Every node in the tree is marked with probability 1/2, and a node is either marked or unmarked. We want to find a marked node and return it. Our algorithm ...
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2k views

Expected number of neighbors

Given a row of 16 houses where 10 are red and 6 are blue, what is the expected number of neigbors of a different color?
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671 views

About a weighted sum of hitting times for random walks on graphs

Consider a random walk on an undirected, non-bipartite graph. Let $\pi$ be the stationary distribution of this process, and let the hitting time $H(s,t)$ be the expected time until a walk beginning at ...
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2answers
1k views

can I get an upper bound on the tail of a binomial variable?

I have $X_1,...$ Bernoulli variables with probability $p$ of success. I want to get an $n$ such that with probability $\delta$ $P(\sum_{i=1}^n X_i \ge k) \ge \delta$ This $n$ would of course depend ...
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Probability of the propagation of a rumor problem

On a small island there are 25 inhabitants. One of these inhabitants, named Jack, starts a rumor which spreads around the isle. Any person who hears the rumor continues spreading it until he ...
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1answer
2k views

Probability of cumulative dice rolls hitting a number

Is there a general formula to determine the probability of unbounded, cumulative dice rolls hitting a specified number? For Example, with a D6 and 14: 5 + 2 + 3 + 4 = 14 : success 1 + 1 + 1 + 6 + ...
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1answer
412 views

Gaussian Distributions and probability mass

Consider n loaded dice with differing probability distributions. I have constructed an equation that tells me the probability mass, the number of times a certain number is expected to be thrown. Now ...
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1answer
97 views

How to use the cumulative dist. fun to verify an exact density (Probability)

I have this table: x | 0 1 2 3 4 5 ------------------------------------------ f(x) | .7 .2 .05 .03 .01 .01 F(x) | .7 .9 .95 .98 .99 1 Where f(...
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1answer
846 views

Probability Theory, Chernoff Bounds, Sum of Independent (but not identically distributed) r.v

Is there any known bound on sum of independent but not identically distributed geometric random variables? I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for ...
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2answers
705 views

Probability of Fire

The probability that a fire will occur is $0.001$. If there is a fire, the amount of damage, $X$, will have a Pareto distribution given by $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An ...
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3answers
961 views

Optimally combining samples to estimate averages

Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-...
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1answer
345 views

Sufficient conditions for convergence of functions of random variables

I hope this question is not too general, but I am not completely sure yet how to phrase it. So we all know that when we have two sequences of random variables $X_n$ and $Y_n$ for $n \ge 1$, that ...
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3answers
217 views

Relationship between these two probability mass functions

If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows: $P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $ what can ...
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0answers
188 views

Covariance and Random Variables

Suppose $X$ and $Y$ are independent random variables. The mgf for $X$ is $M_{X}(t) = (1-t)^{-1}, \ t<1$. The mgf for $Y$ is $M_{Y}(t) = (1-2t)^{-1}, \ t< 0.5$. Two other random variables $U$ and ...
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2answers
827 views

convergence of sequence of random variables

What does this expression mean - $\lim_{n\rightarrow\infty} E|X_n-X|=0$? $X_n$ is a sequence of random variables and $X$ is a random variable. What does this expression imply? Can I say that the ...
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1answer
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How can I prove this law of “generalized” total probability?

Given that $X_i$ is a partition of the probability space, and $Y,Z$ are events in the probability space such that $P[Y \cap X_i] > 0$ for all $i$, how can I prove that $$P[\,Z\;|\;Y\,] = \sum_{i=1}^...
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1answer
253 views

Given probabilities for a branching process, how do I compute the probability mass function at a time?

I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the ...
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1answer
347 views

How can Gröbner bases used to describe discrete probability?

I am working in the field of machine learning, and I have come across a few papers that show relationships between Gröbner bases and discrete probability. So I come here for help. Can you please ...
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3answers
2k views

Price Calculation based on probabilities

This is a strange question, it might be easy for all you math wizards or it even may be impossible. If you don't understand what I mean let me know so I can change the way I post the question. ...
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1answer
2k views

chi-square distribution with $v = 7$ degrees of freedom Ti-89?

I need to figure out the Probability that $Y > 23$ for a chi-square distribution with degrees of freedom $= 7$. The book says to do this with some applet. I don't know what it's talking about. ...
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2answers
176 views

Probability to get always the same number choosing randomly from a set of $c$ elements

What is the probability to get $n$ times the same element $k$ choosing randomly from a set $A$ knowing the cardinality $|A| = c$? Thank you in advance, rubik
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3answers
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Probability concerning unfair dice with N sides

Fair warning: I am not a math expert (that's why I'm here). I would like to be able to calculate the probability of rolling a certain side on a die with n sides where any number of those sides has an ...
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3answers
368 views

Proof of $P(X < x) = F(x−)$

The definition of a Cumulative Distribution Function $(CDF)$ says that $$P(X \le x) = F(x)$$ This is all good. Then my text book gives the following theorem without proof: $$P(X \lt x) = F(x-)$$ ...
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2answers
158 views

Probability and Integrals

Suppose $f(x,y) = c$ for $0\lt y\lt x\lt 1$ and $0$ outside. What is $P(X+Y \leq 1)$? What is $P(X^2+Y^2 \leq 1)$? So \begin{equation*} P(X+Y \leq 1) = \int_{0}^{1} \int_{0}^{1-x} 2 \ dy \ dx? \...
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3answers
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Expected number of (0,1) distributed continuous random variables required to sum upto 1

I define $X_i$ as a random variable that is uniformly distributed between (0,1). What is the expected number of such variables I require to make the sum go just higher than 1. Thanks
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2answers
222 views

claims in an insurance portfolio

Claims arrive in an insurance portfolio, according to a homogenous Poisson process $X(t), t\geq 0$. Assume that each of the $12$ months in a year has exactly $30$ days. Calculate the value of the ...
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2answers
92 views

More choices lesser draw or more draw lesser choices?

I was always thinking about this questions when it comes to lottery. lets say you have a number range from 1 - 45. You need 5-6 numbers to win. Let says you have 2 options Draw 3 sets of numbers with ...
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3answers
947 views

Interpretation of joint pdf/product of marginal pdfs

If we have two Random Variables X and Y, what is the interpretation for the three cases: a] p(X=x,Y=y) = $p_1$(X=x) * $p_2$(Y=y) b] p(X=x,Y=y) < $p_1$(X=x) * $p_2$(Y=y) c] p(X=x,Y=y) > $p_1$(X=x)...
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1answer
113 views

n agents accessing a resource

I want to solve a simple stochastic problem. Imagine there are n agents who want to access a resource, with a probability p at a given time t. What ist the probability that the resource will be free ...
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2answers
500 views

Confidence band for Brownian Motion with uniformly distributed hitting position

Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that $$ P(\...
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1answer
165 views

Apple App Store Review Rate

If 72% percent of new apps are reviewed within the last 7 days, then there is a 72% chance that any app is reviewed in the last week. $$72\% = \sum\limits_{i = 0}^{6} {x\left( {1 - x} \right)^i }$$  ...
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3answers
1k views

How safe is it to ignore low probability events?

See this question on applying probability theory principles in software design. The question is generally the following: you design some system (say software) and rely on some well-known mathematical ...
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1answer
454 views

Normal Distributions and Claim Amounts

Suppose for company A, there is a $60 \%$ chance that no claim is made in the coming year. If one or more claims are made, then the total amount is normally distributed with mean $1000$ and standard ...
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An occupancy problem

Consider the scheme of random placing balls into $N=1000$ cells. We continue the procedure of placing balls as long as a last cell remains empty. The process terminates when a ball is placed into this ...
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3answers
2k views

combination of brownian motion

Suppose $B_t$ is a Brownian motion. As I understand, $B_2-B_1$ is independent of $B_3-B_2$ from properties of Brownian motion. Does it also mean that $B_1$ and $B_2$ are also independent? Can I use ...
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1answer
2k views

lower bound for probability distribution of a random variable

If $X$ is a random variable with finite mean $\mu$ and variance $\sigma^2$, how do I show that the estimate \begin{equation*} P[\mu − d\sigma < X < \mu + d\sigma] ≥ 1 − 1/d^2~\forall d>1 \...
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3answers
228 views

$P(A)$ from $P(B|A)$, $P({\rm not\ }B|{\rm\ not\ }A)$, $P(B)$

If A = { Market research predicates strong demand } and B = { Market demand is strong }, can we reasonably assume that P(A or B) = P(A) * P(B)? The problem is that I know P(B|A) = 0.8 P(not B | ...
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1answer
200 views

Why does this inequality in probability theory hold?

Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that: $$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \...
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1answer
732 views

Concentration of poisson distribution

Let $X$ be a Poisson random variable with mean $\lambda$. How do I show that $P[X \geq \lambda] = 1/2$? Also, I was wondering what distributions have this property that the density is concentrated ...
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2answers
369 views

Expected number of tosses so that 1 out of 3 bins has 2 balls in it

You have 3 bins. You can toss balls one at a time to the bins until any one of them has 2 balls in it, and then you stop. The tosses are independent, and each bin is equally likely to be hit. It's ...
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4answers
8k views

Distribution of Functions of Random Variables

In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For ...
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1answer
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Sweepstakes Probability

Is there a way to determine the probability of winning a particular sweepstakes with only the following information: -the estimated odds of winning are 1 in 13,000 -contestants must be owners (...
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2answers
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Calculating a sample size based on a confidence level

It's been a while since my last statistics class... I have 404 files that went through some automated generation process. I would like to manually verify some of them to make sure that their data is ...
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1answer
301 views

Moment Generating Functions

Suppose one is given an arbitrary moment generating function $M_{X}(t)$. How would you determine $P(X=k)$ from this? We know that $M_{X}(t) = E[e^{tX}]$ and $M_{X}(0) = 1$.
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1answer
233 views

A component with more edges than vertices in a random graph

I am working with $G_{n,p}$ random graphs. Let $p$ (the probability of an edge) be $c/n$, so $c$ is the expected number of neighbors for a vertex when $n$ is large. Now let $N$ be the number of ...
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1answer
717 views

what will be the distribution of ratio of correlated gamma distributed random variables?

If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes ...
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2answers
586 views

What is the probability that every pair of students studies together at some point?

A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 ...
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1answer
122 views

can I bound the following probability?

I have a probability space $\mathcal{X} \times \mathcal{Y}$. $\mathcal{X}$ and $\mathcal{Y}$ could be infinite, but they are at worst case countable. There is a matrix $A_{xy}$ ranging over $\mathcal{...
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1answer
230 views

How long to choose n out of 2n numbers?

Choose numbers from $1$ to $2n$ uniformly at random. How many numbers must be chosen, on average, before at least $n$ numbers have been picked? This is similar to the coupon-collector problem, but ...