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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14 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
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
12 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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4answers
39 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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3answers
403 views
+100

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

Say I have a (fair) die with $F$ faces. I roll it $M$ times. How can I calculate the probability that some face was not seen for $N$ or more in a row of the $M$ rolls? (any face, not just a specific ...
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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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0answers
20 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
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
17 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
24 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
9 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
6 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
44 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
30 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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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
21 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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1answer
31 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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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 ...
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0answers
13 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 ...
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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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2answers
31 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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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
16 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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1answer
269 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
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2answers
29 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 ...
7
votes
1answer
106 views

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) 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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2answers
47 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 ) ...
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2answers
74 views

Probability in multiple choice exams

A multiple choice exam has 175 questions. Each question has 4 possible answers. Only 1 answer out of the 4 possible answers is correct. The pass rate for the exam is 70% (123 questions must be ...
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0answers
9 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a simple lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, ...
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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 ...
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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
295 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
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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1answer
45 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 ...
2
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1answer
40 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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3answers
35 views

Chance of winning simple dice game

Tossing two fair dice, if the sum is 7 or 11, then I win; if the sum is 2, 3 or 12, then I lose; if the sum is one of rest of numbers then I toss the two dices again. What is probability of winning? ...
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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 ...
0
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1answer
37 views

Trying to solve for a variable within a choose function.

I am trying to solve for lowest possible k in the equation: $$1-\frac{{40 \choose 10-k}{10 \choose k}}{50 \choose 10} > .5$$ This comes from a question that I've only been able to brute force ...
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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 ...
2
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3answers
63 views

n distinguishable balls into n boxes

We have n distinguishable balls (say they have different labels or colours). If these balls are dropped at random in n boxes, what is the probability that: 1- No box is empty? 2- Exactly one box is ...
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3answers
85 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ Claim: for any constant $c$, the event ...
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2answers
37 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
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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
27 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
10
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3answers
2k views

On average, how many times must I roll a dice until I get a 6?

On average, how many times must I roll a dice until I get a 6? I got this question from a book called Fifty Challenging Problems in Probability. The answer is 6, and I understand the solution the ...
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4answers
1k views

A plan to defeat a betting game where the odds of winning are 50/50. Help me understand why it's flawed. [duplicate]

My friend has this plan where he implies that it's impossible to lose, as long as the odds of winning are 50/50 on each bet. His idea is that basically you keep doubling your bet until you win and ...