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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12 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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0answers
11 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
16 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 that ...
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0answers
6 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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0answers
20 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
21 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
14 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
4 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
13 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
10 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
26 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
6 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
19 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
19 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
21 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
54 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
25 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
14 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
18 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
26 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)? ...
0
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1answer
10 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
9 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 ...
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1answer
22 views

Probability of Random Event and Conditionality

A company has been running a television advertisement for one of its new products. A survey was conducted. Based on its results, it was concluded that an individual buys the product with probability ...
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1answer
22 views

bernoulli trials and job interviews

I was trying to think of a way to give a hopeful spin to my friend's unsuccessful job interview outcome and I remembered Bernoulli trials which apply to anything with 2 outcomes like "heads or tails" ...
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0answers
17 views

Expected size of largest weakly connected component?

Given an undirected graph of n vertices and n randomly assigned edges, one edge from each vertex, what is the expected size of the largest connected component? For example, with four vertices, there ...
4
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3answers
162 views

Probability of one stock price rising, given probabilities of several prices rising/falling

So this is the problem: An investor is monitoring stocks from Company A and Company B, which each either increase or decrease each day. On a given day, suppose that there is a probability of ...
0
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1answer
10 views

Gamma distribution - closed towards multiplication

First observe how the gamma distribution function can be written in terms of the incomplete gamma function. $\boldsymbol{(1)} \qquad G(y) = \int_{0}^{y} \dfrac{c^{\gamma}}{\Gamma(\gamma)} x^{\gamma - ...
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0answers
18 views

Probability in 3 multiple choice exams [on hold]

a student is taking 3 multiple choice exams in which each question has 4 choices. there are 16 multiple choice questions on each exam and the minimum passing grade is 10 correct questions. What is the ...
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2answers
15 views

introductory probability Q

john is taking a multiple choice test which consists of 8 questions, each question is has 4 possible answers with only one correct. Find the probability that the final answer given is the 6th one that ...
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0answers
14 views

Expected size of set resulting from n random samples with replacement from population of size N [on hold]

If I am sampling n times with replacement from a population of size N, what is the expected size of my resulting sample set? How many distinct elements am I expected to get?
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1answer
18 views

basic probability birthday question

I figure this is a trivial question since it's right in the beginning of the book but I get a different answer from that of the answer in the back of the book. I get .0847 while in the correct answer ...
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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 ...
2
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1answer
46 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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0answers
13 views

How to smooth a probability density function to cover all real numbers

For a personal research project, I have data and its log likelihood $\ell(\theta)$ will depend on the density of the data's distribution. My data is sampled according to a standard sine wave. In other ...
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1answer
28 views

Determining the next observation with a 95% confidence.

Suppose $X$ follows a Poisson distribution with an unknown parameter $\mu$. The outcome of an experiment gave a value $X=625$. I want to determine, given this outcome, the interval in which the next ...
3
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1answer
31 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 ...
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2answers
31 views

Probability of getting 6 letters right

A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 ...
2
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0answers
27 views

probablity - $n$ previolusly persons failed exam

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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2answers
50 views

Tossing two dice with sum equal to 4?

Exercise: Throw two dice. Suppose that eye sum are 4. Calculate the resulting conditional probability that a) the first dice gave a 3 . b ) the second dice gave two or fewer eyes. c ) ...
2
votes
1answer
42 views

$\frac{S_n}{n} \to -1 \ \ a.e.$, exercise from probability book

I'm stuck with this exercise from Williams, probability with martingales. Let $X_1, X_2, \ldots $be independent random variables with $$P(X_n = n^2-1 )= \frac{1}{n^2}$$ $$P(X_n = -1 )= ...
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2answers
15 views

Value of lambda in poisson distribution

I am currently studying statistical estimators and I came across a question that asks to give an estimate of the parameter λ of a Poisson distribution (using the method of moments), given that the ...
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2answers
32 views

Probability of Seven (Distinct) car accidents occurred on the same day

Seven (Distinct) car accidents occurred in a week. What is probability that they all occurred on the same day? My Solution: All 7 accident occurs in 1 day in $\binom{7}{1}$ ways All 7 ...
1
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0answers
26 views

trying to understand binomial distrubition

I'm trying to understand when I can use the binomial distribution. I have searched some examples online and I'm wondering if I can use them in this situation: if we had a deck of 20 cards and we ...
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1answer
33 views

Indicator function property

The indicator function for a probability event $A \subset \Omega$ is given by $ \mathbf{1}_A(x) =\begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A. \end{cases}$ Consider $N$ ...
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0answers
9 views

the covariance matrix of the joint distribution of two dependent random vectors.

Let the vectors $x$ and $y$ be jointly normally distributed with mean $\mu$ and covariance matrix $V$. Let $x \sim \mathcal{N}(\mu, \Sigma)$ and let the nonrandom matrix R have full column rank. What ...
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1answer
46 views

Exercise from Williams book Probability with martingales

I'm doing this exercise from Williams book Probability with martingales Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n ...
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0answers
17 views

Does a state which is passed at least 3 times had to be passed 5 times in Markov chain

Prove of disprove: Let $\{X_n\}_n$ be homogenous Markov chain. if we start from state $i$, there is a positive probability that we pass at least 3 times at state $j$. Does it follows that exists ...
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0answers
13 views

Find UMVUE for $X_1 \dots X_n \sim N(\theta, 1)$ [on hold]

Let $X_1, \dots, X_n \sim N(\theta, 1)$ with $\theta$ being the parameter we are trying to find. The question I am working on says as a hint to proving that $\bar{X}$ is the UMVUE, we should first ...