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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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25 views

$X_n$ converges in probability but not almost surely

Consider the sequence $\{X_n\}$ given in Davide Giraudo's answer to this question. As is explained in the answer, the lack of a.s. convergence comes from the Borel-Cantelli lemma. Another way of ...
1
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2answers
68 views

Find the pdf , distribution function of $X$ and $E[(X-2)^2]$

I 'll be very grateful if you can help me , here is the question : When a person sends an email, the probability that there is an attachment is 0.5. If there is an attachment then the size of the ...
-1
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1answer
32 views

I need help with calculating probability [duplicate]

I need help with this question: Consider a loaded dice such that the probability to obtain an outcome of 1 is 2p/3, the probability of obtaining 2, 3, 4 or 5 is p each, and the probability of ...
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0answers
8 views

Distribution of vector of distinct elements

Assume we have a set of numbers $[S] = \{1,2,\dotsc,S\}$ and we pick uniformly at random $L$ elements $\mathbf x = (x_1,x_2,\dotsc,x_L) \in [S]^L$ from this set (with repetitions). Let vector $\mathbf ...
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1answer
37 views

Calculating probability of broken machine using Bayes Theorem

I am getting really confused with the probability problem below. I found the answer to part $(a)$ is $0.0469$ using the Bayes Theorem, but I am not sure if this is right. An industrial machine ...
6
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1answer
94 views
+50

Difference between Bernoulli random variables

Given are $n$ independent Bernoulli random variables with parameters $p_1,\dots,p_n$. We want to split them into two parts so as to minimize the expectation $\mathbb{E}[|X-Y|]$, where $X$ is the sum ...
0
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1answer
47 views

A group of $200$ persons consisting of $100$ men and $100$ women is randomly divided into $100$ pairs of $2$ each

A group of $200$ persons consisting of $100$ men and $100$ women is randomly divided into $100$ pairs of $2$ each.Find the maximum chance that at most $30$ of these pairs will consist of a man and a ...
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1answer
22 views

“Normalized” covariance matrix of a Gaussian random vector

Let $X\sim\mathcal{N}(0,I_{d})$. I would like to compute the the following quantity: \begin{equation} \mathbb{E}\bigg[\frac{XX^{\top}}{\|X\|_{2}^{2}}\bigg]. \end{equation} Letting $B=\frac{XX^{\top}}{...
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0answers
22 views

Escape time probability distribution

I have a system where a random walker is moving on $\mathbb{Z}$. However, at each point in $\mathbb{Z}$, there is a probability $q$ that an escape route exists along which the walker can escape. I ...
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0answers
34 views

Variance of sum of correlated variables

I want to compute the variance of this estimator $\hat{\sigma}^2 = \frac{n}{N}\sum_{i=1}^{N}\big(R_{i} - \frac{1}{N}\sum_{j=1}^{N}R_{j}\big)^2$, where $R_{1}, \ldots, R_{N}$ are i.i.d such that: $ R_{...
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1answer
26 views

finding some value for a martingale

I got $(X_n)$ being iid random variables. $P(X_n=0)=1/2, P(X_n=1)=1/3, P(X_n=-1)=1/6.$ and $S_n = X_1 + ... + X_n$ I wanna find $\theta \in (0,1)$ that $M_n = \theta^{S_n}$ is a martingale w.r.t. ...
2
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2answers
60 views

What is the probability that $\exists N \in \mathbb{N}$ such that $\forall n > N$, $2n \in C + C$?

Suppose $C$ is a random subset of $\mathbb{N}\setminus\{1, 2\}$, such that $\forall n \in \mathbb{N}\setminus\{1, 2\}$, $P(n \in C) = \frac{1}{\ln(n)}$ and the events of different numbers belonging to ...
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2answers
25 views

Domain of definition of sum of random variables

If $X$ is a random variable over the sample space $S_1$ and $Y$ is a random variable over the sample space $S_2$, then how can we define the domain sample space of the random variable $X+Y$. please ...
2
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1answer
31 views

How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that $Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite ...
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0answers
16 views

How much time would be needed If the bitcoin network were used to find a GUID collision in a database with 10 million guids?

Although calculating hashes is not the same as generating a GUID, it would be insightful to understand how big(or small) a Guid is in comparison to current peer to peer calculating power https://www....
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1answer
21 views

Probability Density Function without having the definite integral [on hold]

So we're given a random variable $X$ and an $f(x) =\begin{cases} 0, & \text{if $x<0$} \\ cxe^{-x}, & \text{if $x≥0 $} \end{cases}$, where $c$ is a constant. How do I find the constant if ...
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0answers
19 views

Statistics probability wording problem [duplicate]

The question I am trying to solve goes like this: The books in a bookstore is known to have one typo per page. In this bookstore, a statistics book has 300 pages. What is the probability the ...
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0answers
19 views

How to define a function whose expectation operator changes with a variable?

To elaborate, I have a function, $C\left(a,b,r\right)$, where, $a,b \in \left[0,1\right]$ and $r \in \left[r_0,r_1,r_2,r_3\right]$. Also, $C_{0,0} = \displaystyle{\min_{r}}~ C\left(0,0,r\right)$, ...
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2answers
26 views

Using P(A'|B') to find P(A∪B)

We have P(B) = 3/5 and P(A'|B') = 1/3. I started with P(A∩B)' = P(B)' ∙ P(A'|B') = 2/5 ∙ 1/3 = 2/15. Then converted P(A∩B)' to P(A'∪B'). Is P(A'∪B') = P(A∪B)', meaning we get P(A∪B) = 1 - 2/15 = 13/...
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1answer
21 views

How do you validate a pmf? [on hold]

$$P\left(X=x\right)=\left\{ \begin{array}{rl} 0.02 & X=0\\ 0.03 & X=1\\ 0.2 & X\in\left\{ 2,\,5\right\} \\ 0.25 & X\in\left\{ 3,\,4\right\} \\ c & X=6 \end{array}\right.$$ I have ...
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1answer
27 views

Intersection with ${x > 30}$ and ${x > 10}$

I am doing a question of conditional probability. Why does: $$P[x > 30 | x > 10] = \frac{p[x>30] \cap p[x>10]}{p[x>10]} = \frac{p[x>30]}{p[x>10]}$$ and not: $$P[x > 30 | x &...
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2answers
45 views

Consider the standard Brownian motion ${W(t), t\ge 0}$: find $p(W(1)\ge 0, W(2)\ge 0)$? [on hold]

Consider the standard Brownian motion ${W(t), t\ge 0}$: find $p(W(1)\ge 0, W(2)\ge 0)$ ?
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0answers
19 views

Distribution for a sequence of dependant variables

Let’s say I have a sequence of numbers: (1, 1.5, 2, 2.5, 4, 6) Which is sampled from some distribution. Let’s say that the numbers are not independent; I know a priori that the interval between the ...
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0answers
38 views

Show that if $X$ is continuous then the probability space is continuous

Show that if $X$ is a continuous r.v. defined in a probability space, then that probability space is continuous. My proof: Justifying by contradiction, suppose $X$ is continuous in a discrete ...
0
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0answers
25 views

Quick terminology question - “Standardizing” Random Variables

This is a really boring question I apologize. I'm not sure what to call this process, I call it "standardization" however perhaps non beginners would think I'm being sloppy with terminology. For ...
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0answers
27 views

Show that all functions $X : \Omega \longrightarrow \mathbb{R}$ defined in a probability space is a random variable.

Show that all functions $X : \Omega \longrightarrow \mathbb{R}$ defined in a discrete probability space is a random variable. $\Omega$ is a sample space Answer: Because the definition of a random ...
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1answer
52 views

The probability of king of spades in a two card draw

Given a standard deck of cards our goal is to pick the King of Spades to win and we draw two cards in each round. What would the probability of winning given these two cases: 1. We draw the first ...
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0answers
16 views

ranking unrelated items

I am trying to rank some related/unrelated items and asking for your help in the identification of a solution. Let assume that we ask a person whether he likes fruits and vegetables, and he replied ...
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0answers
26 views

Proof of mutually satisfactory bet.

Suppose that two people, $J$ and $K$, desire to make a bet. Person $J$ will pay 1 dollar to person $K$ if a specific event $A$ occurs, and a person $K$ will pay $x$ dollars to person $J$ if the event $...
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0answers
29 views

Gaussian mixture model: sum of 2 random variables [closed]

We have data for $X$ (integers). Model $X = Y+Z$. Where $Y\sim N(\mu_1,\sigma_1)$ and $Z\sim N(\mu_2,\sigma_2)$ are latent variables. How to use a Gaussian mixture model or EM algorithm to estimate:...
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0answers
46 views

Map a random variable to a Gaussian

If I have a random variable $X \in \mathbb{R}^n$, under which conditions is there a $C^1$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\varphi(X) \sim \mathcal{N}(0, I_n)$ (...
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1answer
30 views

Questions relating to choosing a j-tuple (j≤n) of positive integers whose sum is at most n.

Let j and n be positive integers, with j ≤ n. An experiment consists of choosing, at random, a j-tuple of positive integers whose sum is at most n. a) Find the size of the sample space. Hint: ...
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0answers
24 views

Is there some sort of a closed formula for $\alpha_n$?

Suppose $\{X_i\}_{i = 0}^\infty$ is a Galton-Watson branching process ($\{X_i\}_{i = 0}^\infty$ is a sequence of random variables satisfying the conditions $P(X_0)= 1$ and $P(X_i) = \Sigma_{i = 1}^{X_{...
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1answer
44 views

Where am I wrong in my solution?

A probability question reads, "A box contains 4 balls. The color of each of the balls is one of the three: White, Black or Red. However, you don't know how many balls of each color are there. It ...
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1answer
16 views

Convergence of sequence of independent random variables

Let $x_n$ be a sequence of independent random variables such that $P(x_n = 0) = 1 - \frac{1}{n}$ 1) Does $x_n$ converge to $0$ almost surely 2) Does $x_n$ converge to $0$ in probability ...
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2answers
47 views

Expected number of turns so that all balls get painted in blue color

A container contains $5$ red balls. On each turn,one of the balls is selected at random,painted blue,and returned to the container.The expected number of turns it will take before all $5$ balls are ...
1
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1answer
22 views

Probability based on a percentage

We have a group of 15 people, 7 men and 8 women. Randomly selecting 5 people, what's the probability to pick 3 men and 2 women? What's the probability to pick at least 1 man? I tried solving the ...
0
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1answer
15 views

How to create a covariance between two distributions?

I have two distributions $A$ and $B$ that are i.i.d. I want to create two distributions $A'$ and $B'$, that have the "same distribution" as $A$ and $B$ (meaning the same probability distribution ...
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1answer
17 views

Convergence almost surely of $a_nX$

Suppose $X$ is a random variable and $\{a_n\}$ is a real sequence converging to 0 as $n\to \infty$. It's clear to me that if $\sum_{i=0}^\infty a_i^2<\infty$ then the sequence formed by $a_nX$ ...
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1answer
65 views

Markov chain with known duration

Let's say there are 2 states (A & B), where the probability of going from state A to B in interval i is $P_{ab}$. State B always lasts for n intervals, then always goes to back to state A. I ...
4
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1answer
35 views

Two boys pick a subset of $40$ toys that they like. They can pick the same ones. What is the probability that they picked three same toys or more?

Two boys pick a subset of $40$ toys that they like. They can pick the same ones. What is the probability that they picked three same toys or more? My answer would be $$\frac{ \sum_{ i =3}^{40} \binom{...
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1answer
20 views

Do we need a sigma algebra to define finitely additive probability measures

In order to define a (countably additive) probability, we need a $\sigma$-algebra. I'm interested in finitely (but not countably) additive probability measures. My questions are: (1) Do we need a $\...
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2answers
66 views

Probability that of having exactly two boys

If in a family of 3 children, there is at least 1 boy, what is the probability that there are exactly 2 boys among the children? My attempt: I calculated all the possibilities of the family by ...
0
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0answers
19 views

Example of higher random vector moments

While reading about random vectors, I learned that... $$ E\left[\vec{X}\right] = \left[\begin{array}{cccc} E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\...
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0answers
24 views

Proof in Variational Gaussian Approximation

Here is the result from the appendix of paper The Variational Gaussian Approximation Revisited (Manfred Opper and Cedric Archambeau) This result has been used in the popular paper Practical ...
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1answer
23 views

Chromatic number of a subgraph of a random graph

Suppose that we have a random graph G(n,p) with $n$ vertices and each edge exists with probability $p = n^{-\alpha}, \alpha>\frac{5}{6}$. Prove that with high probability, say $1-\delta$, every ...
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0answers
25 views

Probability of drawing higher sum? [closed]

An opponent and I draw four cards from a deck. What is the probability that the sum of my cards is higher than my opponent's?
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0answers
19 views

Order statistics with uniform distribution [closed]

Let $X_1, X_2, X_3, X_4$ be i.i.d random variables from a uniform distribution over the interval ($\theta$, 8), where $\theta$ is a constant smaller than 8. Let Y = min$\{X_1, X_2, X_3, X_4 \}$. In ...
6
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2answers
141 views

How much should I pay for a chance to win 100$?

There are $4$ closed doors, with $100\$$ behind one of them. You can pay $X$ to open a door. If the money is there, you can keep it. If not you can pay another $X$ to open the next door, and so on. ...
1
vote
1answer
22 views

Operational Meaning of Relative Entropy

Is there an operational meaning to understand the non-negativity of relative entropy between two probability distributions? I understand the mathematical argument/proof. But I want to know if there is ...