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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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let X,Y be independent poisson distributed random variables with parameter alpha and beta respectively. E(XY)?

E(XY) = double summation of (XY)*f(X,Y), but I don´t have the f(X,Y). X+Y would be Poisson distributed with parameter alpha + beta...but what about XY? Not sure what else to do. Thanks in advance.
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7 views

Strong markov property and another stopping time

I'm trying to prove that given a regular continuous time markov chain $X_t$ (pure jump process) its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous markov chain, where $T_n$ is the time of the $...
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Markov Chain holding time

Let $X$ be a continuous-time Markov chain. How does one justify $P(X(s)=x,0\leq s\leq t\mid X(0)=x)=\lim_{n\to\infty}P(X(kt/n)=x,k=0,1,\dots,n\mid X(0)=x)$ without prior knowledge of the ...
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1answer
14 views

Calculate probability of union of events

Let $A$ and $B$ be events in sample space $S$ such that $P(A) = \frac{1}{2}$ and $P(A' \cap B') = \frac{1}{3}$. Find $ P(A\cup B')$. I found that $P(A\cup B) = \frac{2}{3}$ and $P(A' \cap B)$ [that ...
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0answers
5 views

Is entropy function lipschitz? [on hold]

Is the information entropy function Lipschitz with respect to $l_{1}$ norm or $l_{2}$ norm?
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17 views

Probability that 5 students seated in a row are reseated such that none have the previous nieghbours

There are five students $S_1, S_2, S_3, S_4 , S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4, R_5$ arranged in a row initially $S_i$ sits on $R_i$. For $j = 1, 2, 3, 4$ ...
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0answers
9 views

Token Bucket Algorithm

Can the token bucket algorithm solve the below problem? Say I'm selling a product online and announce this to 1 million people. Assume incoming traffic will look similar to a Poisson distribution. ...
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0answers
10 views

Tricky information inequality $I(X;Z) \geq H(T)$

I am wondering whether $I(X; Z) \geq H(T)$ when the following conditions hold: $H(T | X) = H(T)$ $H(T | Y) = H(T)$ $H(T | X, Y) = 0$ $H(Y | Z) = H(T | Z) = 0$ $X, Y, Z, T$ are discrete. I know first ...
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1answer
24 views

Probability drawing balls out of an urn

Seven balls are randomly withdrawn from an urn that contains $12$ red, $16$ blue, and $18$ green balls. Find the probability that $3$ red, $2$ blue, and $2$ green balls are withdrawn. The answer is $...
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2answers
17 views

Counting sample points dice experiment

Suppose that die have been altered so that the faces are $1,2,3,4,5,5$. If the die is tossed five times, what is the probability that the numbers recorded are $1,2,3,4,$ and $5$ in any order? This ...
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0answers
19 views

Hard demonstration - brilliant minds. I came up with the idea of ​obtaining the limit distribution of an estimator [on hold]

I do not know exactly how to get there, I thought to use the estimate of maximum likelihood but I'm not sure... If $X_1,\ldots , X_n$ are i.i.d. according to the uniform distribution ${\cal U} (0, \...
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10 views

difficult demonstration, Convergence in distribution of subsamples? [on hold]

I am trying to solve this convergence but I am not know exactly what is the n1, please give me some ideas: Let $X_1$,...$X_n$ i.i.d with cdf $F(\varepsilon_p)$=$p$ and $f(\varepsilon_p)>0 $ if $\...
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1answer
8 views

What is the average number of matches when randomly picking letters

Suppose we have three pieces of paper. On the first one you have the letter A, on the second on the letter B, and on the third one the letter C. Now suppose I'm going to randomly pick each one from a ...
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2answers
20 views

Mean stopping time of a Brownian motion

I came across the following proof of the fact that the mean stopping time of a Brownian motion to hit $-1$ or $1$ is $1$: Let $B$ be a Brownian motion. We already know $B_t^2-t$ is a martingale. Let $...
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1answer
11 views

Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
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1answer
14 views

Geometric Probability. Sphere

Suppose a sphere with radius $R$. Find the probability of the event such that $n$ selected points of the sphere are within the distance of $r = \frac{R}{2}$ from the center of the sphere. The points ...
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17 views

probability of picking same ball in n trials

Here is my problem: I have t unique hidden balls in an urn I must pick k balls each draw each time I draw a ball I can uncover it. I have n tries; in each try(draw) I have all t balls available ...
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1answer
20 views

What is the PMF of the product of two discrete random varibales? [on hold]

Let $X$ , $Y$ be two discrete random variables. What is the probability mass function of $Z=X Y$?
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1answer
15 views

Complicated Demonstration - Violation of the theorem that converges in probability and not in distribution

I was thinking that if a sequence of random variables $Y_n$ with c.d.f. $H_n$ which converges to $c$ in probability, such that $H_n(c)$ does not converge to $H(c)=1$. How could I make an example ...
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2answers
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Compute Probability with random draws without replacement for certain order

I would like to know if the solution to the problem is correct. If not, could you please explain the reasoning? There are 7 white balls and 6 black balls in a bag. We draw without replacement. What ...
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0answers
29 views

In an experiment of rolling a die $5$ times, event $A$ is considered success if you get number 5 or number 6.

In an experiment of rolling a die $5$ times, event $A$ is considered success if you get number 5 or number 6. Calculate: a) Probability that event $A$ is success $4$ times. b) Probability of not ...
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0answers
13 views

Rounding numbers in statistics or probability calculations for discrete real-world examples (chi-squared test)

I wanted to ask about rounding numbers in statistics and probability calculations for discrete real-world values. Let's give an example which interests me – it can be applied to many other examples. ...
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0answers
17 views

Lognormal distributed random variable excercise

Let Y be a random variable distributed Lognormally, Q1 be it's first quartile, Q3 it's third quartile and M be its median: prove that- M - Q1 < Q3 - M I have managed to figure out that Q3 > M > ...
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2answers
31 views

Probability of being matched up in chess tournament

The chess clubs of two schools consist of, respectively, $8$ and $9$ players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players ...
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2answers
19 views

Probability of choosing 2 items of the same color from a bucket with k colors

I think this is right, just want to double check. Let's say I have a bucket of n items. Those n items have k colors uniformly distributed. Then the probability that I get two items from the same color ...
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1answer
20 views

Conditional distribution for two random variables

I recently came across and exercise from a past exam and I was wondering how to solve it. Two independent random variables $A$ and $B$ are given and they both follow the exponential distribution but ...
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0answers
27 views

Probability problem about a parking lot

We want to design a parking lot for a group of 200 apartments still under construction. It is known that for each department (from city statistics) the number of cars will be 0, 1 and 2 with ...
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2answers
18 views

If $X$ is a r.v. is there an independent r.v. $Y$ on the same space?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $X,Y$ two random variable with distribution $F_X$ and $F_Y$. I know how to construct a probability space $(\Omega ',\mathcal F',\mathbb P')$...
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0answers
17 views

How to calculate pi sub D, upper opt

I¡m learning Bayesian Networks and now I'm studying Decision Making. I have the following probability table (sorry about the format, I don't know how to do it better): ________| $\psi(y,t,d)$ $+y,+t,...
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23 views

How to find the probability of escaping rooms? [on hold]

Suppose you are playing an escape game with $9$ rooms in succession, i.e. you must escape the 1st room to get to the 2nd room, and so on. If you fail to escape a room in the allotted time, the game ...
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0answers
16 views

calculate the probability of error in a array of bits

I need to calculate the probability in a certain problem. So there are 555 random bits [1 0 1 0 ... 1 0 0 1 1]. These 555 bits are divided in 37 parts of 15 bits each. Of the 555 bits, X bits flip at ...
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0answers
9 views

When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
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1answer
26 views

How to find equation for $\theta_\lambda$

Suppose we have data $D=\{x_i,y_i\}_{i=1}^n$ where $x=(x_{i,1},x_{i,2},1)^T \in \mathbb{R}^3$. An estimator for these data, $$y=f(x;\theta)=(\theta_1 x_1+\theta_2 x_2+\theta_3) ~~~(\theta \in \mathbb{...
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0answers
31 views

Let $X, Y$ be random variables which are not independtent and whose distribution we know. How can we find distribution of $XY$?

Let $X, Y$ be random variables which are not independent and whose distribution we know. How can we find distribution of $XY$?
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Martingales - cyclists

The cycling race, in which $100$ cyclists participate, is played according to the following rules: after the k-th stage all riders who managed to fit in the top ten at all k in stages receive $10^k$ $....
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0answers
26 views

Two independent geometric variables [duplicate]

Suppose that $X$ and $Y$ are independent, identically distributed, geometric random variables with parameter $p.$ Prove that $P( X = i | X + Y = n) = 1/(n-1)$ for $i = 1,2,\dots ,n-1.$ So I said $$P(...
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1answer
18 views

I'm playing a video game where I got an item with 1/150 odds from 3 crates. The 3rd crate gave me 2 of the item. What are the odds of this happening?

The problem is noteworthy because the game is supposed to stop giving you this item after you have 3 of them, so I might be one of the only people in game with 4 of these. Relevant details: The item ...
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0answers
18 views

How to find Random variable without replacement sample on pmf [on hold]

How to find another random variable on PMF in without replacement sample
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0answers
16 views

Covariance of two random variables, linear relationship and normalization of covariance

The covariance of two random variables $X$ and $Y$ is given by $$\displaystyle\operatorname{cov}\left[X,Y\right]=\mathbb{E}\left[\left(X-\mathbb{E}\left[X\right]\right)\left(Y-\mathbb{E}\left[Y\right]\...
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0answers
45 views

If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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1answer
35 views

What is the probability mass function of X

I'm having trouble with the following question: Initially we have 1 red and 1 green ball in a box. At each step we choose a ball randomly from the box, look at its color and then in put it back into ...
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1answer
33 views

Probability of singletons in with an uncountable sample space

Let us assume that we have a sample space $\Omega=[0,1]$. Why is it not possible to have all the singletons $\{x\}\in\Omega$ with non-zero probability measure. I searched for an answer on this forum ...
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1answer
30 views

On the convergence in probability of a sequence of random variables.

Let $\{X_t \}_{t \in \mathbb{N}}$ be a sequence of independent random variables such that $E[X_t] = \theta E[X_{t-1}]$ for all $t \in \mathbb{N}$ where $|\theta|< 1$ and $E[X_0] = \mu > 0$. ...
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0answers
18 views

justification of $\sum_{j\neq i}\mathbb{E}[Y_i Y_j] = (n - 1)\mu^2$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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1answer
10 views

Given distributions of 2 indpendent variables, find the probabiliy distribution of a function of the 2 independent random variables? [on hold]

Given $f_{X}$ and $f_{Y}$ how do you find the distribution $f_{g(X,Y)}$ of a function $g : \mathbb{R}^{2} \rightarrow \mathbb{R}$?
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2answers
25 views

which rule or definition apply ${\displaystyle {\begin{aligned} \operatorname {E} [Y_{i}^{2}] = (\sigma ^{2}+\mu ^{2}) \quad (3.1) \end{aligned}}}$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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0answers
35 views

Is every probability measure a Radon measure? [on hold]

Let $\mu$ be a probability measure defined on a compact convex subset $K$ of a locally convex Hausdorff space $X$. Is $\mu$ a Radon measure?
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1answer
46 views

Inserting random numbers from 1 to $n^2$ in a matrix of size $n \times n$

I have two matrices of size nxn with random numbers that are in range of $1$ to $n^2$. I'm trying to calculate the probability of : the numbers 1 and 9 are present in the same indices in the two ...
2
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1answer
37 views

What's the probability that in a box of a dozen donuts with flavors randomly picked out of 14 no more than 3 flavors are in the box? (Check my work)

Suppose that in a donut shop that offers 14 flavors of donuts there's a "grab bag" box with random flavors thrown in, each flavor equally likely for each donut. What is the probability that the box ...