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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Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
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0answers
13 views

Probabilistic Logic

I was wondering if there is any system of logic that has been worked out that explicitly uses probabilistic notions at its foundation. It would include ideas like as a first principle, all statements ...
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2answers
22 views

Determing a transition probability matrix

I need some support with this homework exercise: An urn contains at most $N$ balls. Let $X_n$ be the number of balls in the urn after the $n$-th execution of the following procedure: If the urn is not ...
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1answer
20 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
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21 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
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18 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
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1answer
16 views

independence and characteristic functions [duplicate]

Why is it that \begin{equation*} \mathbf{E} [e^{i t_1 X_1} e^{i t_2 X_2}] =\mathbf{E} [e^{i t_1 X_1}]\mathbf{E} [e^{i t_2 X_2}] \end{equation*} for RVs $X_1, X_2$ and all $t_1, t_2\in\mathbb{R}$ ...
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1answer
17 views

probability density and distribution functions

I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$. 1) Calculate the density functions, distribution function and characteristic function of $C = ...
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0answers
10 views

Confidence interval of exponential random variables

I have a sequence of random variables $X_1, X_2, ..., X_n$ such that $X_i = e^{-(x_i-Θ)}$ I have to construct a confidence interval of the form $[Θ−c,Θ]$,where $Θ = \min _i{X_i}$. For $n = 10$ how ...
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7 views

Galton Watson process

Let $X_n$ the number of individus of the $n^{th}$ generation. For example suppose that a father has no brother and sister and does $3$ children. Suppose that thefather is the generation $0$ (i.e. ...
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2answers
24 views

What's the chance of $(\frac{1}{2})^x$ with $y$ iterations?

If I have a program that creates, let's say, one billion integers, with each having a pure $50 - 50$ chance to be one or zero, what is the chance of finding $x$ zeros in a row? for brownie points, ...
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0answers
13 views

Theoretical probability that everyone in the U.S. is separated by 6 degrees [on hold]

The six-degrees-of-separation theory says that I can be most certain that I have a friend who has a friend, who has a friend, who has a friend, who has a friend, who has a friend, who is friends with ...
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0answers
13 views

Convercenge in probability implies convergence in Lp [on hold]

Show that if $X_n$ is that $|X_n|< C$, with $C\in \mathbb{R}$, $\forall n \in \mathbb{N}$, then $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
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1answer
12 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
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1answer
15 views

Confidence Interval Question - Steps Taken, no given standard deviation

I just wanted to make sure I was doing this Confidence Interval problem correctly (or incorrectly). Q: The following are the daily number of steps taken by a certain individual in 20 weekdays. (some ...
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1answer
34 views

Probability in a Restaurant

In a revolving restaurant, there are four round tables each with three seats. How many different ways can $12$ people sit in this restaurant? This is what I think the answer is: $$\binom{12}{4} ...
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2answers
59 views

What are the odds of any role of a 24 sided die occurring 4 or more times in 10 rolls?

Note that I am not asking about the odds of a chosen roll happening 4 times in 10 rolls, (this has a probability of 0.000517 according to a binomial calculator), rather, the odds of ANY roll happening ...
1
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1answer
17 views

Hypergeometric function variance

In a fishing event, a small lake is populated with $75$ trout, among which $25$ are tagged. Each participant is allowed to capture $5$ fish during the day (the fish are not put back into the lake). ...
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1answer
20 views

Two related question, in one. Same topic: Dispersion..

$1.$ Prove: If $X_1,X_2,X_3,\ldots,X_n$ are independent random variables then: $$D\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n D(X_i)$$ Proof: Because of independence we have: $$D(\sum_{i=1}^n ...
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0answers
16 views

Proving that each element in reservoir have equal probability of been selected in reservoir sampling?

Here is the description of the algorithm and proof of the correctness The algorithm creates a "reservoir" array of size $k$ and populates it with the first $k$ items of $S$. It then iterates through ...
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0answers
24 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
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0answers
11 views

How to recompute the markov transition matrix given a reduction to the number of states? Clustering from a transistion matrix

I am been puzzled with this one for sometime. Given a transition matrix (as below) for a markov chain of N states; how do we calculate the transition matrix for N-1 states, where we combined stat n1 ...
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1answer
27 views

What is the probability of picking two black cards out of a pack of ten?

I have ten cards; eight of them are red, and the remaining two are black. What is the probability of choosing both black cards in four draws? I have tried $\frac{3 \cdot 4}{2} \cdot \frac{3 \cdot 3 ...
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48 views

combinatorics contest problem

Question: Calvin has a bag containing $50$ red balls, $50$ blue balls, and $30$ yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out $5$ more red ...
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1answer
19 views

A die is thrown $n$ times. $X_1$-number of times a number from $\{1,2,3\}$…

.. $X_2$ number of numbers that fell from $\{4,5\}$, $X_3$ number of $6's$ that fell. Find $$P\{ X_1=k\mid X_2=m\};0\leq m \leq n.$$ Now, I believe that $X_3$ is completely irrelevant here. What I ...
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0answers
8 views

Probability in Entity Linking

This question is about computer science probability, in particular Natural Language Processing, but I think that there is a little too much math in order to ask it on stackoverflow. Anyway, I'm ...
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2answers
21 views

In a box which has balls numbered 1..100 , 5 balls are drawn.

$X$- random variable that represents the largest number of the 5 drawn. Find the distribution of $X$. Now, it seems that this random variable is of discrete type. What I have trouble it defining it ...
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1answer
33 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
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3answers
34 views

probability about coins [on hold]

A gambler has two coins in his pocket, a fair coin and a two-headed coin. He picks one at random from his pocket, flips it and gets heads. What is the probability that he flipped the fair coin?
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25 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
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2answers
25 views

Distances to the center of points uniformly distributed in a disk

We choose $n$ points at random from the surface of disk of radius $1$ (the points are chosen with equal probability). If we omit the point furthest from the center (from $n$ points), what is the ...
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6 views

Maximize the net profit with probabilities — optimal purchasing

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
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1answer
17 views

Find expected value of random process [on hold]

I found a problem, which I can't solve: Let's say that $\tau \sim Unif(0,1)$ distribution. Suppose that $X_t=(1\!\!1_{[0,\tau]}(t))^2, t\in[0,1]$. What is the $EX_t$ and $var(X_t)$? I don't know ...
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0answers
105 views

Pareto distribution,fisher information, confidence interval

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
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1answer
24 views

Need help with probability homework [on hold]

Alright so I could use some help with my homework, thank you in advance! Plura goes to the gym 15% of the days of the year. Carla goes to the gym 20% of the days of the year. a) what is the ...
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30 views

Expectation problems in probability.

Dan tosses a coin $n$ times independently, while the probability for a unique tail is $1\over 3$. For $1\le k\le n$, let us denote the number of sub-sequence of length $k$ of H's. For example, if n=5 ...
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2answers
25 views

During a night, each chameleon changes its colour to one of the other four colours with equal probability.

Five chameleons of all different colours meet one evening. During the night, each chameleon changes its colour to one of the other four colours with equal probability. Find the probability that the ...
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2answers
21 views

Counting problem: ways of opening stores in non-adjacent blocks?

A coffee company wants to set up stores along the main street of town, which has $n$ blocks. The company won’t open two stores in the same block, or in two adjacent blocks. Q: For this coffee shop, ...
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2answers
38 views

how to solve this conditional probability

Manufacture A and B produce one type of electrical element, given that the probability of produced element being faulty is $0.05$ for A and $0.01$ for B. If two of these elements has been picked, from ...
3
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1answer
30 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
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1answer
26 views

Probability of co-occurence

Of total $N$ people, $m$ people are good at mathematics and $c$ people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...
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1answer
23 views

proof of the convergence of confidence intervals

The confidence interval can be derived intuitively by replacing the standardized mean with the standard normal and variance with sample variance, but is there a formal limit? I'm trying to prove if ...
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4answers
174 views

Find the probability that the final score is 4 in a dice game with two throws

A game uses an unbiased die with faces numbered 1 to 6. The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then die is thrown again and the ...
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0answers
33 views

Probabilities: Meeting people

The probability of women meeting a man is $m$. Let's look at the perspective of a specific man. The probability of meeting him is $\tilde m$. Say women look twice for men. Then (assuming $\tilde m$ ...
2
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1answer
22 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
2
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0answers
27 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
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1answer
22 views

Derivative of a CDF [on hold]

Suppose that $X$ is a random variable whose mean is $m$. I need to show that $\frac{\partial}{\partial m} \text{Prob}\{X\geq x\} >0$. Intuitively, increasing the mean I'm shifting probability ...
3
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3answers
30 views

Property of cumulative distribution function

I was taking the course on random variables , where I faced below property of cumulative distribution function: $$\lim_{x\rightarrow a^+}F_X(x)=F_X(a^+)=F_X(a)\qquad\qquad ...
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Prove: $E[E(X|X+Y))=E(X) ; E[X|X+Y] = n/(n+m)*(X+Y)$ [duplicate]

E[E(X|X+Y)]=E(X) is diferent of E[X|X+Y]. And in E[X|X+Y] I give the final result that is "n/(n+m)(X+Y)", and I am asking to demonstrate that E[X|X+Y] = n/(n+m)(X+Y). Let X and Y be independent and ...
1
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1answer
30 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...