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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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 ...
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3answers
16 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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30 views

Probability of choosing a point from infinite 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,... $ For a value $ m $, what is the probability of selecting a point $ p(m,y) $ so that $ ...
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0answers
22 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 ...
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0answers
19 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, ...
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3answers
32 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. ...
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1answer
17 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 ...
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0answers
17 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

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?
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1answer
22 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 ...
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0answers
25 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 ...
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0answers
18 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 ...
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0answers
17 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 ...
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1answer
28 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 ...
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0answers
17 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 ...
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1answer
22 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}[\ ...
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0answers
25 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 ...
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3answers
39 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 ...
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2answers
19 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 ...
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3answers
61 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 ...
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1answer
13 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? ...
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1answer
13 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: $$\text{E}[X] = \int_{0}^{\infty}x f_{X}(x) dx $$ $$ = \int_{0}^{\infty}\int_0^x dy f_{X}(x) dx $$ ...
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3answers
49 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 ...
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1answer
26 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 ...
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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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4answers
351 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
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4answers
23 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 ...
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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 ...
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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)
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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 ...
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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
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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 ...
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1answer
36 views

an exercise related central limit theorem

I'm working on the following problem in Durrett: Let $X_1, X_2, ...$ be i.i.d, nonnegative, $EX_i=1$ and $Var(X_i)=\sigma ^2$. Then we have $2(\sqrt{S_n}-\sqrt{n})$ converge to $\sigma \chi$ in ...
3
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1answer
32 views

Limit (in probability) of sequence of independent random variables

We have $\{X_n\}$ independent random variables which converge to $X$ in probability. I was asked to prove that $X$ is constant. My approach is to try to show that$Var(X)=0 \implies X$ constant, but i ...
3
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2answers
45 views

Markov Chains - Strong Markov Property

I'm struck with an exercise. I tried, but the results don't seem to fit to those proposed. Exercise: Two players play the following game. The one who begins draws two cards from a deck of 40 cards ...
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1answer
34 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
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0answers
11 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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1answer
22 views

Law of Iterated Expectation with Probability?

I'm trying to follow a proof of the following proposition (source) Let X and Y be two independent random variables and denote by $F_X(x)$ and $F_Y(y)$ their distribution functions. Let $$Z=X+Y$$ ...
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3answers
63 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
1
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2answers
44 views

Showing that infinite product of random variables goes to zero: $\prod^\infty X_i \rightarrow 0 \text{ a.s.}$

I am doing the following exercise: Let $X$ be a strictly positive rv with $\mathbb E[X]=1$ but $X \neq 1$ almost surely. Let $X_1, X_2 \dots$ be iid with same distribution as $X$. Now let $M_0=1$ and ...
3
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3answers
69 views

How biased is this biased coin

Suppose that we have a coin that we suspect is biased, but that we don't know precisely how biased it is: all we know is that its probability p of landing heads is some fixed value between .4 and .6, ...
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0answers
23 views

How to solve following sequence equation?

Suppose I have a sequence ${p^*_j}$, $j=0,...,m$, satisfying relations $p^*_j=p^*_{j-1}p_{j-1,j}+p^*_{j}p_{jj}+p^*_{j+1}p_{j+1,j}$, with \begin{equation} p_{jj}=\frac{2j(m-j)}{m^2}, \end{equation} ...
1
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1answer
22 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
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3answers
33 views

probability of exactly one out of N events occuring

I have N events. Each "i" event has probability $P_i$. What is the probability of $n$ events occuring? I have seen this answered for two and three events, but not for an arbitrary N. In principle, ...
1
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1answer
34 views

Roulette with p=$\frac{2}{3}$. What is the probability of not going home?

I'm learning about the gamblers ruin. The problem is that I don't know how to calclate the formula. I got two exercise questions in my book. Both of the questions will be about a strange roulette ...
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1answer
22 views

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ ...
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0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
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1answer
20 views

Condition of reversibility of Markov Chain [on hold]

Show that a Markov Chain is time reversible iff $\pi _{i} P_{ij}= \pi _{j} P_{j i}$
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1answer
25 views

Why do all steady state probabilities have the same denominator?

I have noted that the steady state probabilities of an irreducible Markov chain can be written as fractions that have the same denominator. Is there any result about this property? What does this ...
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1answer
42 views

Probability of winning consecutively [on hold]

India and USA play $7$ football matches. No match ends in a draw. Both the countries are of same strength. Find the probability that India wins at least $3$ consecutive matches.