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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1
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2answers
69 views

Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
1
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0answers
26 views

Differences of Markov chain is Markov

In my studies of Markov chains, I was tackled with this tough problem: Let $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain with transition probabilities satisfying $ | i-j | > 1 \to ...
0
votes
0answers
14 views

Forecasting for a non-steady state problem

Let say, at each discrete time step, $t_i$, we can forecast of a specific event's occurring rate, $I_{trans}$ by following formulation: $$I_{trans}(t) = I_{ss}. \sqrt{(\frac{A\tau}{t})} .e^{-\frac{B\...
8
votes
1answer
69 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
-6
votes
1answer
35 views

Finding the number of possible shortest ways. [on hold]

Find the number of possible shortest ways from A to B.
0
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3answers
39 views

Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
2
votes
1answer
35 views

Arrange 18 pips on a die with at least one 0 side to maximize the probability that 5 rolls sum to 13 or more.

You are arranging pips on a standard 6-sided dice. Rules: At least one side must be left blank at 0. The average roll must be 3 (so, you have 18 pips to distribute among five sides). You want to ...
0
votes
2answers
24 views

Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
4
votes
1answer
61 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
0
votes
0answers
10 views

Drift analysis of an absorbing Markov chain

Consider a set $S$, and suppose we have a sequence of random subsets $$ \zeta_t = \{x_1, \dots, x_n\} $$ for $x_1, \dots, x_n \in S$. We do not know with which probability density the points of each $\...
0
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0answers
46 views

Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
1
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2answers
51 views

Find the probabilty of 25 random people

X is the weight of one person, $X \sim N(\mu =78,\sigma =13.15 )$. If I choose randomly 25 people, what is the probability that the average of their weights will be $86$ ? I define $\displaystyle Z =...
0
votes
1answer
28 views

Conditional probability by joint probability

I have the joint pdf $$f(x,y)=\frac{1}{5}(3xy^2+2x^3y);0<x<1;0<y<2$$ and I have to calculate $$P(\frac{1}{2}<Y|X<\frac{1}{2})$$ I have found that $$f_{X}(x)=\int_{0}^{2}\frac{1}{5}(...
0
votes
1answer
28 views

Are $X_1$ and $X_2$ independent?

Let $X=(X_1,X_2)$ be an absolute continues random vector with the density function $f_X(x_1,x_2) = \left\{ \begin{array}{ll} \frac{2}{3}x_1+\frac{4}{3}x_1 x_2+\frac{2}{3}x_2, & \mbox{for } (...
0
votes
3answers
55 views

True or false:if $A\subset B$, then $P(A)<P(B)$?

They ask me if this statement is true or false, and explain why. They suggest I write an example showing why it is false or true. The statement is: if $A\subset B$, then $P(A)<P(B)$. What I ...
2
votes
2answers
69 views

If we've got 10 coupons, what is expected number of different ones if there are 25 different types

I can't figure out this problem : There are 25 different types of coupon, all equally probable to get. If we have got 10 coupons, what is expected number of different coupons between them? ...
0
votes
0answers
21 views

Bounding Probability Distribution

I have the following problem. Let $X$ be a continuous random variable with image $[0, b]$ for some finite $b>0$. So we have finite moments, $\mathbb{E}[X^n]$. I am hoping to say something about the ...
6
votes
3answers
88 views

Probability of choosing $n$ numbers from $\{1, \dots, 2n\}$ so that $n$ is 3rd in size

We uniformly randomly choose $n$ numbers out of $2n$ numbers from the group $\{1, \dots, 2n\}$ so that order matters and repetitions are allowed. What is the probability that $n$ is the $3^{\text{rd}}$...
0
votes
0answers
13 views

Is there any example of a Markov chain (discrete) with limit distribution (discrete) of heavy tail?

Is there any example of a Markov chain with limit distribution (discrete) of heavy tail? In other words, a Markov chain whose limit distribution has infinite second moment?Already, thanks for the help!...
2
votes
1answer
49 views

Which has higher variance, coin toss vs dice roll?

Dusting off some high school stats and getting confused over the following: Two betting games: Pick right side of coin, even-money bet ($p = 0.5$, $q= 0.5$), Pick right value in a 10-sided ...
1
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0answers
38 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
1
vote
1answer
53 views

How to choose between two options with a biased coin

We would like to choose between theatre and cinema by tossing a coin. Unfortunately the only available coin we have has probapility of heads $p\ \left(\dfrac{1}{2}<p<1\right)$. How could we use ...
0
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0answers
39 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
-1
votes
0answers
15 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
2
votes
1answer
18 views

The intuition behind conditional probability and independence in the case of different sample space

I came up with this question when doing this problem: In throwing a pair of dice, let A be the event that "the first die turns up odd", B the event that "the second die turns up odd", and C the ...
0
votes
0answers
18 views

Data transmission process PDF

Given the quasi-defined data transmission random process: $X(t) =\sum_{n=-\infty}^{+\infty} a_n \pi_T(t - nT)$ where $a_n$ are statistically independent RVs that can either assume the value 0 or 1 ...
0
votes
1answer
27 views

Probability in the game Resistance

I was playing the game Resistance with a group of 10 people. In the game, people are given one of two "assignments". 6 people are given cards that tell them they are part of the Resistance. 4 people ...
5
votes
1answer
45 views

Difficulty understanding step in Kac's proof of Feynman-Kac Theorem

I am trying to understand a proof of the Feynman-Kac Theorem, as set out in Mark Kac's 1949 paper 'On Distributions of Certain Wiener Functionals'. Kac defines a series of independent and ...
2
votes
2answers
28 views

References for the applications of probability in gambling

The intuition behind many theorems in probability comes from gamblers' games. I would like to know if there are any books or articles which cover some such connections between probability and its ...
0
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2answers
33 views

Is there a name for the distribution of this CDF function?

CDF: $F(x) = (1-e^{-a \cdot x^2})^{\frac{b}{c-x}}$ where $a,b,c$ are positive constants, and $x \geq 0$. Can any body give some advice on how to analyze the mean, variance or any other properties ...
1
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0answers
28 views

Game theory: how is law of large number applied here?

This is a claim rephrased and lifted from from Herbert Gintis' book "Game Theory Evolving" Pg187 Consider an evolutionary game with $n$ pure strategies $i = \{1, \ldots, n\}$, and time periods $t ...
-1
votes
1answer
33 views

does this converge? [on hold]

If I have $$X_n=\begin{cases}x_n & p_n\\ 0 & 1-p_n \end{cases}$$ and I know that $x_n$ converges to $0$ as $n$ tends to $0$, can I say that $X_n$ converges to $0$ almost sure?
1
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2answers
30 views

Permutations in an Infinite List of Random Numbers

In an infinite list of random numbers from a to b, prove that in this list, there are all possible permutations of n numbers from the list, where n can be any number. Here are some versions of the ...
-1
votes
2answers
67 views

Expected number of tosses to get 3 consecutive Heads [on hold]

I have a fair coin. What is the expected number of tosses to get three Heads in a row? I have looked at similar past questions such as Expected Number of Coin Tosses to Get Five Consecutive Heads ...
1
vote
1answer
18 views

Finding the method of moments estimator for the Uniform Distribution

Let $X_1, \ldots, X_n \sim \text{Uniform}(a,b)$ where $a$ and $b$ are unknown paramaters and $a < b$. (a) Find the method of moments estimators for $a$ and $b$. (b) Find the MLE $\hat{a}$ ...
1
vote
1answer
25 views

Betting ended after nth round.Find the sum of money NOT WON?

Rahul and Vijay are playing a game with 12-sided die,where both of them lay bets on outcomes of roll of die.They start betting Rs 5 each on first round of the game and the amount bet in each ...
1
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0answers
27 views

Calculate probability of an event occurring exactly once in an arbitrarily selected year, given mean annual occurrences over a century

If an event occurs an average of 0.6 times/year over a century, what is the probability that it occurs exactly once in a randomly selected year? I was able to find p(occurs exactly once in a given ...
-1
votes
0answers
27 views

conditional expected value and not mutual indipendent events [closed]

$\newcommand{\P}{\mathbb{P}}$ Let be $E,G,H$ pairwise independent events but not mutual (e.g. $P(E\cap H)=\P(E)\mathbb{P}(H),\,\P(G\cap H)=\P(G)\P(H), ...but \,\P(E\cap G\cap H)\ne\P(E\cap G)\P(H)$ ...
-4
votes
0answers
29 views

$\newcommand{\P}{\mathbb{P}}$ If a,b are independent, can I prove that $\P(ab|c)=\P(a|c)\P(b|c)$ and $\P(a|bc)=\P(a|c)$? [closed]

$\newcommand{\P}{\mathbb{P}}$ If a,b are independent, can I prove something below: A. $\P(ab|c)=\P(a|c)\P(b|c) $ B. $\P(a|bc)=\P(a|c)$ thanks
5
votes
3answers
134 views

Is there a quick way to justify that this elementary probability is equal to $2/3$?

I just solved this problem with the conditional probability formula and after a while the answer was surprisingly $2/3$. I believe there must be a tricky short way to calculate it. Can somebody help ...
0
votes
1answer
26 views

Prove $X$ and $Y$ are not independent

Let $X$ and $Y$ be two random variables. Their joint probability density function is $$f: (x, y) \mapsto C(y^2 - x^2)e^{-y} \mathbf{1}_A(x, y)$$ where $C \in \mathbb{R}$, $A = \{(x,y) \in \mathbb{R},...
0
votes
1answer
39 views

Does this converge?

If I have $$X_i=\begin{cases}2\quad p=\frac{1}{3}\\ \frac{1}{2}\quad p=\frac{2}{3} \end{cases}$$ random variables with the same distribution. How can I compute the limit almost sure as $n\to\infty$ ...
0
votes
1answer
36 views

5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : ...
3
votes
1answer
27 views

Figuring out probability of dice with least amount of questions

Given $n$ dice, each with $k$ faces numbered $1,\dots,k$, you're allowed to ask me what the probability of some event happening is (a subset of all the possibilities and I give a number). What ...
1
vote
2answers
40 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
0
votes
1answer
32 views

Probability: Application Of “Expected Value”

$\newcommand{\P}{\mathbb{P}}$$\newcommand{\E}{\mathbb{E}}$So, I was learning expected value today and I'm trying to understand the significance of calculating this term "Expected value". In this ...
1
vote
1answer
34 views

If the diameters of ball bearings are normally distributed, determine the percentage with diameters between $0.610$ and $0.618$ inches.

If the diameters of ball bearings are normally distributed with mean $0.6140$ inches and standard deviation $0.0025$ inches, determine the percentage of ball bearings with diameters Between $0.610$ ...
1
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0answers
35 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
-3
votes
0answers
20 views

The odds of an event when time is a constant [closed]

A YouTube video is 100 minutes long. In the comment section someone has placed a quote from the video. What are the odds of someone else reading this quote at the exact same time the quote is being ...
0
votes
1answer
23 views

What is the difference between a reversible markov chain and a reversible in equilibrium markov chain?

In the text I'm using it says: Let X = {$X_n : 0 \leq n \leq N$} be an irreducible Markov chain such that $X_n$ has the stationary distribution $\pi$ for all $n$. The chain is called reversible if ...