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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6 views

Markov random walk never hitting zero (and probability axioms?)

Consider a simple Markov random walk on the integer state space {0, 1, 2, …}. Every time slot in which we are not in state 0, we move left with probability $\theta$ and right with probability ...
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1answer
36 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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1answer
13 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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1answer
24 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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3answers
447 views
+100

Probability of some die face being missed N or more times in a row in M rolls?

Please see edits below - it appears my use of "run" is an abuse of terminology and was confusing. Markus says it most precisely in our conversation (perhaps replace "each" with "some specified"): ...
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0answers
40 views

probablity - $n$ previolusly persons failed exam ( extremely difficult)

We have an exam. Students are staying in queue. After every student probablity that professor finish exam is $\frac12$. For first student in queue there is $\frac12$ probablity that he pass. When ...
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0answers
5 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
12 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
26 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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0answers
16 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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1answer
19 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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2answers
47 views

$X_n \to X$ in $L_2$, show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$

$X,X_1,...$ are random variables, $X_n \to X$ in $L_2$. Show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$. My attempt: $X_n \to X$ in $L_2 \implies \lim_{n \to \infty} E[(X_n-X)^2]=0 \implies \lim_{n ...
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1answer
29 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 ...
2
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0answers
14 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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0answers
14 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 ...
1
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1answer
20 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
27 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 ...
1
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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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2answers
21 views

How do I determine the proper probability for a chance-based reward system?

Let's say I have game of chance with a number of players (1000), with each player having a chance to win something (\$25 for this example.) What probability to win would each player need in order to ...
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0answers
19 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
32 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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1answer
34 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 ...
2
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1answer
276 views

Distribution function (CDF) of the sum of two random variables + law of iterated expectations

I'm taking my first probability class, and we're studying sums of independent random variables. We're using Ross's First Course in Probability. It states the definition of a convolution, but doesn't ...
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0answers
11 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, nor ...
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0answers
10 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 ...
2
votes
1answer
19 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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4answers
58 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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2answers
24 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, ...
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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
33 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
5 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
24 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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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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3answers
116 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
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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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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
7 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
28 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
22 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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0answers
27 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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1answer
25 views

Uniform distribution over an hyper-ellipsoid

Let $\mathbf{X} \in \bf{R}^p$ be a random vector whose elements are uniformly distributed over the hyper-ellipsoid $x^TAx<1$, (where $A$ is a positive-definite matrix). Is it possible to compute ...
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0answers
16 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 ...
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0answers
19 views

Question with the value at risk (VaR) criterion

Let $X$ and $Y$ be the random payoffs from two different investment strategies. Recall that the Value at Risk (VaR) criterion with parameter $\gamma \in (0,1)$ decides $X \succ Y$ if and only if ...
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1answer
28 views

standard deck and the probability of at least one card,exactly one void and two voids

The question is this: if 13 cards are dealt from a standard deck of 52, what is the probability that these 13 cards include a)at least 1 card from each suit b) exactly 1 void(e.g no clubs)? ...
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1answer
11 views

mutually exclusive events where one event occurs before the other

This question has been asked before. Here is the link: Mutually exclusive events Here is the description to the problem: Let E and F be mutually exclusive events in the sample space of an ...
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0answers
11 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
2
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1answer
48 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
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1answer
44 views

In how many of the possible arrangements will both end balls be of the same colour?

Suppose 6 blue balls, 4 red balls, and 2 white balls are placed in a straight line. In how many of the possible arrangements will both end balls be of the same colour? This is a self-answered ...
3
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1answer
32 views

Playing the St. Petersburg Lottery until I lose everything

This question continues the following question: Calculating the probability of winning at least $128$ dollars in a lottery St. Petersburg Paradox Here is a lottery: A fair coin is flipped repeatedly ...