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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3 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
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1answer
9 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
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1answer
20 views

Probability of a nonnegative submartingale converging to zero

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot.
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1answer
10 views

In Markov chains a limit distribution is invariant

Suppose we have a Markov chain $(X_n)_{n \geq 0}$ with state space $S$. Suppose that $(\pi_i)_{i \in S}$ is a limit distribution. Then is $(\pi_i)_{i \in S}$ an invariant distribution ? I know the ...
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2answers
25 views

Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
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3answers
26 views

Probability of number of people in car park at any given time

A building has 22 car spaces, each having a car parked within each spot in the morning. Each car is retrieved by its respective owner at some point (random time) between 7am and 9am (120minutes). Each ...
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1answer
13 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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0answers
8 views

Equation to denote a set based on probabilities

I have a set R with elements r. Each element has a certain probability P(r|X). Now a want a formal equation/notation for a new set E which contains the expected r elements when X happens. I can't ...
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2answers
14 views

How to calculate the probability of a die with a wild side?

So I have a 6-sided die with 5 different values in 5 of their sides. Its sixth side can be treated as any of the other 5 values. So my question revolves around which is the probability of getting any ...
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2answers
19 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
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1answer
3 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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0answers
15 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a sequence of natural numbers (i.e. this is a probability space of sequence of natural numbers sometimes used in ...
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0answers
12 views

Struggling with the notation of conditional expectation

Here's the question. I know the fomula of $E(X|Y=y)$ where y is a paticular number. But for this question, I really don't understand what's $E(X|Z)$ meaning. Help!
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0answers
16 views

Probability 2 balls in a bag [on hold]

consider an experience which consisit of drawing 2 balls with replacement from an urn containing 7 balls of which 3 are blue and 4 are yellow (i)what is the sample space (ii)define the events as a ...
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1answer
34 views

Find the value of $a$ and $b$ in $ F(x) = a + b \arcsin x $

Given $X$ is a continuous random variable and its probability distribution function is $$F(x)= \begin{cases} 0, & x < -1, \\ a+b\arcsin x, & -1 \le x < 1, \\ 1, & x \ge 1 ...
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1answer
21 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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0answers
7 views

MATLAB code based on a binomial random matrix [on hold]

I have a matrix, A, with $40000$ binomial random elements. I am trying to complete the following code and would appreciate help: I need to create $40000$ arrays $X_n$, where $X_n$ represents the ...
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1answer
22 views

Let n>=2, k>=2. The set of all k-element subsets partitioned into 4 classes: (i) class of subsets containing both 1 & 2, how many k-element subsets?

Sorry for the long title, I'm new here & not sure of the appropriate way to post long questions. The full question is: Let n>=2,k>=2. The set of all k-element subsets of [n] may be partitioned ...
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0answers
8 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
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1answer
14 views

Poisson Process problem, transform the possibility notation

Question: Suppose that a store opens at 0 pm and customers arrive according to a non-homogeneous poisson process ${N(t),t\ge0}$ with the intensity function $\lambda(t)=2t+1$ per hour. Let $S_3$ denote ...
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1answer
24 views

Multnomial coefficient combinatorics problem

The following problem: Ten diplomatic delegates are seated in a row. There are two specific seating requirements: 1) France and Britain are sat next to each other, and 2) the U.S. and Russia are ...
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0answers
16 views

How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
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1answer
30 views

What is the probability that a random K-bit odd-number is prime?

Is it $e/K$? In an experiment that created 1000 random RSA-2048 key-pairs, 2000 random 1024-bit primes were created. It turned out that $727,709$ random candidates were generated, to create 2000 ...
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0answers
19 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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1answer
21 views

Challenging Problem of Linear Permutation by H.C. Rajpoot

How many numbers are lying between 20045757087 & 87050752074 when all the 11-digit significant numbers, formed by permuting the digits 0, 0, 0, 2, 4, 5, 5, 7, 7, 7, 8 together, are arranged in ...
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1answer
28 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
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1answer
21 views

combinatrix & probabilities

probabilities have always been something tough to comprehend for me, may be someone can help me on this. So here's the problem: Bob tosses a coin but can't see the result, his friend John can see it, ...
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1answer
30 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
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1answer
13 views

Does the statistical frequency of patterns manipulate the probability of a given event? [on hold]

This is a question I've encountered when I first read about the Gambler’s Fallacy, I'm really wondering why it's considered fallacious? Taking statistics into consideration, If you studied the results ...
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0answers
20 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
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1answer
27 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
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1answer
34 views

Conditional expectation of $X$ given $Z$, where $Z = 1$ if $X > Y$ and $-1$, otherwise

Let $X\sim\operatorname{Exp}(1)$ and $Y\sim\operatorname{Exp}(2)$ be independent random variables. Define $Z$ by $$ Z = \begin{cases} 1,& X>Y\\ -1,& X\leqslant Y. \end{cases} $$ I want to ...
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2answers
46 views

Expected value problem on dice reroll

The question is here: Roll N* 3-sided dice(0,0,1), roll them twice and choose a better result, what is the expected value? If possible I would also like an answer for dice {0,1,2} or {1,2,3} if ...
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0answers
12 views

Entropy of the Random Energy Model

I need to show that $$\text{lim}_{N \to \infty}\frac{1}{N}\text{log}\mathcal{N}(\epsilon, \epsilon + \delta) = \text{sup}_{x \in [\epsilon, \epsilon + \delta]}s_a(x).$$ We have that ...
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1answer
35 views

Lottery winning

This is a ratter simple probabilistic problem but i have not seen any similar. My local lottery works like this: There are 48 numbers in total (numbered from 1 to 48) You have to pick 5 numbers from ...
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2answers
26 views

How many 5-element subsets of [10] contain at least one of the members of [3]?

Here [10] denotes the set {1,2,3,4,5,6,7,8,9,10} & in the same manner [3] denotes {1,2,3}. I'm attempting to solve this for my combinatorics course. My method would be to solve 10 permutation 5, ...
2
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1answer
23 views

Monty Hall problem with pre-specified probabilites

Suppose that a player is given the probabilities for a prize behind each of the three doors. $p_1$, the probability of the prize being behind door 1, is $p_1=\frac{1}{2}$, the other probabilities are ...
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0answers
7 views

Distribution of the sample mean of correlated exponential random variables

My question is how to determine the PDF of $X = \frac{1}{N}\displaystyle\sum_{k=1}^N \frac{X_k}{(X_k + a)^2}$ where $X_k$ are dependently, identically exponential random variables with mean $\lambda = ...
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1answer
17 views

Find the probability generating function of $2X$.

If $X$ follows a poisson distribution with parameter $\lambda$ (mean). Then find the probability generating function of $2X$. I'm getting stuck with forming the expression, as I'm getting confused ...
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2answers
14 views

Let $X$ be a Random Variable. Define $2X$.

I would like to know what exactly the changes are in the values the random variable($2X$) can take, if for example $X$ follows a Poisson or Binomial Distribution. If suppose $X$ follows a Poisson ...
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0answers
12 views

Probability of lead Between two Candidates

Suppose in an election cadidate A receives n votes and cadidate B receives m votes $m<n$.If all orderings are equally likely what is the probability that A throughout leads B?I think the number of ...
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0answers
12 views

Probability of rest of votes, when some votes are already counted

Say, for example, that we had $n$ people voting YES or NO and we have already counted some amount $d$ of the votes and of those $r$ have been YES's. How does this effect (or does it) the distribution ...
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2answers
34 views

How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
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0answers
8 views

Drift of Brownian motion conditioned on Hitting Time

Suppose we have a Brownian motion started from height b>0, with constant negative drift $\lambda$. We can 'calculate' the drift in the following seemingly ridiculous way. We condition on the first ...
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0answers
32 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
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0answers
23 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
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0answers
24 views

Probability of collecting all the sticker types

This question is in the context of tuning a training procedure, whereby the learner may receive random stickers for good performance. I am trying to figure out the probability of any given learner ...
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4answers
68 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
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0answers
30 views

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$?

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$? $$$$ I have tried to use Jacobian matrix to do ...
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0answers
17 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...