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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8 views

This random variable converges in distribution?

$\delta_x$ is a Borel probability that $\delta_x(x)=1$ and $\mu_n$ is a uniform distribution in this interval $(1, 1 + \frac{1}{n})$ The variable $X_n\sim \frac{1}{2}\delta_{\frac{1}{n}} + ...
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votes
1answer
12 views

identifying sudden change in value given a list of values over time

I have a list of the average price of an item in a game over time. Things don't tend to move much. I am wondering how I can detect whether a new value inserted is a surprising movement in price. I ...
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0answers
8 views

Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
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1answer
17 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
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0answers
6 views

how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
2
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0answers
19 views

Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
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0answers
12 views

Comparing Two Ways of Scoring Data

I have used two different methods to give a score to the same data set. One is a discrete method and the other continuous. How do I show that the continuous method is more sensitive to changes in ...
0
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1answer
29 views

Variance of absolute value sums of random variables

Let $X=\left|\sum _{i=1}^n Z_{i} \right|$ and $Y=\sum _{i=1}^n |Z_{i} |$ where random variables $(\textit{Z${}_{i}$})$ are i.i.d, and $Z_{i} =0$,$+1$ or$-1$, with probability ${p}{}_{0}$, ...
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0answers
16 views

Proving that the Poisson compound process has independet increments

Let $X_t=\sum_{i=1}^{N_t}J_i$ be a compound Poisson Process, where $J_i$ are independent and equidistributed. I have to prove that for every $0<t_1<t_2 \leq t_3<t_4$ : $X_{t_4}-X_{t_3}$ is ...
1
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1answer
22 views

Borel-Cantelli question

If $X_1...X_n$ are i.i.d. and $\mathcal{N}(0,1)$ how can Borel-Cantelli lemma helps us to proof a.s. of: $$\max\{X_{n^2+1},X_{n^2+2},\dots,X_{n^2+2n}\}\ge5 \text{, }\forall n>N$$ Thank's for your ...
2
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0answers
16 views

Queue theory - M/D/k - Probability of never having a queue before a time T

This is probably a known result, but I couldn't find any resource pointing directly to the issue I'm trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a ...
1
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0answers
18 views

construction of a path of quadratic variation

Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= ...
0
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0answers
23 views

Distribution of the minimum of two exponential random variables

$X$ and $Y$ are two exponential random variables with rate 1 and 2. lets define random variable $Z$ such that: $z_i = min(x_i,y_i)$, where $i =1,2,3,...N$. Let $V$ be another random variable and ...
0
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2answers
42 views

Find the probability of opening all the boxes

Suppose there are $20$ boxes which $1-20$ are printed on each box. There is a key in each box which are also marked with $1-20$. So only the key with the same number with the box can open it. For ...
0
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1answer
12 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
0
votes
1answer
17 views

how to find the cumulative density function

Consider $$f(x)=3x^{-4} \qquad \mbox{on} \qquad x\geq 1.$$ Let $X$ be a continuous random variable on $x\geq 1$. Find the cumulative distribution $F(x)$ for $X$. I know that CDF for a continuous ...
25
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7answers
4k views

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...
0
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0answers
26 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
2
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3answers
27 views

The non-uniform probability of sums from the throw of multiple dice

I'm reading The Drunkards Walk by Leonard Mlodinow. In the book, the author writes: From a throw of three dice, a sum of 9 and 10 can be constructed in an equal combinations. However, the outcome ...
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0answers
56 views

Probability of choosing a point from large set

Let x and y be non-negative integers and $y \le x \le m$. Let us define a function $ f(x) = x/n, n = 1,2,3,...,m $ For a value $ m $, what is the probability of selecting a point $ p(m,y) $ so that $ ...
1
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0answers
30 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{\frac{5}{9}}$ in distribution, where $Z_n ...
0
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0answers
23 views

Find/estimate variance

Let $w_{11},\ldots , w_{nm}\in [0, 1]$ be a set of constants and $H_1(t), \ldots , H_m(t)$ be some cumulative distribution functions (CDFs). Consider a sample of independent random variables $\xi _1, ...
0
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3answers
39 views

You roll 3 six-sided dice. What is the probability that the third is at least as high as the highest of the previous two?

I know that the probability of the first two dice being different is $\frac56$, and the first/second being greater is $\frac56$, but am not sure how to calculate the prob of the 3rd being greatest. ...
0
votes
1answer
20 views

Cumulative Distribution Funciton to pmf

I am still quite new to cdf and pmf. When we only have pmf for x = 1, 2 and 4 , how should I understand the corresponding cdf as in the pmf for x = 3 doesn't exist. Also I tried to draw the piecewise ...
0
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0answers
25 views

Calculating expectation of random vector

Let $\Omega=\Theta\times \Pi$ be a finite sample space and $P$ be prob. measure on $2^\Omega$. We define random vector $X: \Omega \rightarrow \mathbb{R}^n.$ How can I calculate following conditional ...
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1answer
32 views

Unfair coin tossed twice [on hold]

An unfair coin is tossed twice. The probability of heads is 3 times the probability of tails. What is the probability that at least one head is flipped?
0
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1answer
25 views

independence of random variable

Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables. 1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are ...
1
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0answers
30 views

Notation in probability theory: conditional on multiple events or joint of event with an conditional one

It might be a quite dumb question and if so, I apologize in advance (I am kind of a newbie in probability theory ). But once in a while it bothers me and I can't find the answer to it. Ok, now the ...
1
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0answers
21 views

Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
0
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0answers
18 views

Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
2
votes
1answer
32 views

Expected value of room enters

I was looking at previous exam questions, but one of the questions I don't know how to solve correctly. In this question I need to calculate the expected amount of rooms the mouse enters before he is ...
0
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0answers
18 views

Bayesian Updating - plug in previous posterior for prior?

Let's say I have two sequences of observations, $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$. For each sequence I'm going to estimate the probabilities of certain events occurring, namely event $A$ in ...
1
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1answer
23 views

Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
0
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0answers
29 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
0
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3answers
40 views

Can you simplify this expression?

This is a Bayes formula incorporating 2 random variables. The final expression seems a bit tricky to simplify the exponents and I'm still not so confident with my algebra (pardon me ;)). Can you have ...
0
votes
2answers
22 views

Probability of two strings being equal

Given a matrix $A\in F_2^{n\times m}$, (let $m< n$ and $A$ has full column rank) what is the probability under the distribution ( $y,y'$ uniformly random in $\{0,1\}^m$), such that $Ay=y'$? I am ...
0
votes
2answers
32 views

Tabulate the probability distribution of $x$.

If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? Tabulate the probability distribution ...
1
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3answers
62 views

Sample space: What's the possibility that a family has n boys?

What's the sample space in these two cases? Case1: Of all families with two children you ask the parents, if they have a boy born on a Thursday. They say yes. What's the possibility that the family ...
0
votes
1answer
14 views

What's the probability, and how to choose the right formula?

Question 1: Toss a coin 4 times. Let $A$ denote the event that a head is obtained on the first toss, and let $B$ denote the event that a head is obtained on the fourth toss. Is $A \cap B$ empty? ...
0
votes
1answer
14 views

How to prove that expectation is the integral of survival function? [duplicate]

I am trying to prove that $E[X] = \int_0^{\infty} P (X > x)$ I have started like below: $$\text{E}[X] = \int_{0}^{\infty}x f_{X}(x) dx $$ $$ = \int_{0}^{\infty}\int_0^x dy f_{X}(x) dx $$ ...
2
votes
3answers
59 views

If a die is thrown thrice. Find the probability that the largest score is three times the smallest.

I have no idea about the answer, but I'm viewing the question this way; If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be ...
1
vote
1answer
27 views

How to calculate the probability of a random variable given two independent variables?

For example, to calculate $\mathbb{P}(C \mid X_1,X_2)$, I know $\mathbb{P}(C \mid X_1)$ and $\mathbb{P}(C \mid X_2)$. $X_1$ and $X_2$ are independent random variables. If there is a way to calculate ...
0
votes
1answer
13 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
6
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5answers
363 views

Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
2
votes
4answers
24 views

conditional probability of several events

I'm having a hard time understanding what this question wants: A person initially purchases either type A or type B. She will choose either type A or type B with an equal probability on her first ...
0
votes
2answers
36 views

Calculating the Variance of a Dice Roll?

Here's my thinking: $$Var(X) = E(X^2) - E(X)^2$$ Assuming each roll is independent: $$E(X^2) = E(XX) = E(X) \cdot E(X) = E(X)^2$$ Thus: $$Var(X) = 0$$ However, this is not correct. Where did I ...
1
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1answer
26 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
1
vote
1answer
21 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
1
vote
1answer
19 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
1
vote
1answer
17 views

Notation and a problem with Aleatory variables (Advanced Probability)

I am studying advanced probability and I have a question with notation. One exercise says: Let $(\Omega,B)$, show that $A \in B$ iff $1_A \in B$. But, $1_A$ is a function, what the book means with ...