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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48 views

How can I derive this equation?

This equation is from this paper and I can't understand how the right-hand can be derived from the left-hand using Bayes' rule: $$\frac{p(O_{fg} \mid I, I_t)}{p(O_{bg} \mid I, I_t)} = \frac{p(I \mid ...
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1answer
27 views

Expectation of a transformed normal CDF [duplicate]

If we define a random variable $X \sim N(0,1)$ with $\Phi$ being the cdf of a standard normal, what would $E(\Phi(a+bX))$ be? I was only able to rewrite $\Phi(a+bX)$ as $P(Z\leq aX+b|X)$ with $Z\sim ...
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0answers
46 views

How to determine the values of $w$ when calculating the PDF of $W = X+Y+Z$?

I am trying to solve the problems about finding $X+Y+Z$ where $X, Y, Z$ are uniformly distributed random variables, each in the interval of $[0,1]$. (please reference: Finding the distribution of the ...
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0answers
13 views

How to find the probability (a) of rolling a number (b) on a (c) sided die at least (d) times when rolling the die (e) amount of times? [closed]

How to find the probability (a) of rolling a number (b) on a (c) sided die at least (d) times when rolling the die (e) amount of times? Example: what is the likelihood of rolling a 6 at least 3 times ...
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1answer
31 views

coin toss and conditional probability

A random experiment consists of tossing a coin, followed by more coin tosses depending on the results of the first toss. The coin is fair, but it is deemed to possibly come up "Heads," Tails," or "...
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1answer
28 views

How to reform the Density Functions to make logical predictions in this problem?

I would like your opinion on the following problem which I'm trying to tackle. A boat's radio can be heard from a distance and depending on this distance, the noise of the sensed signal varies. Let's ...
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1answer
12 views

Conditional probabilty joint distribution [closed]

How do you find a conditional distribution of a discrete random variable which is in the form P(Y | X = x), e.g P(Y | X=5)? In my experience both discrete random variables have a value assigned to ...
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1answer
37 views

Let $A_1$, $A_2$, $\ldots$ be an arbitrary sequence of events from a probability space. Define a new sequence of events $B_1$, $B_2$, $\ldots$:

Let $A_1,A_2,\ldots$ be an arbitrary sequence of events from a probability space $(\Omega, \mathcal F, \mathbb P)$. Define a sequence of events $B_1,B_2,\ldots\subset\mathcal F$ as follows: \begin{...
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2answers
51 views

Is it true that $|\mathbb E(XY)| \leq \sqrt{\mathbb V(X)\mathbb V(Y)}$? [closed]

Let $X,Y$ be to random variables, where $\mathbb E(X^2)< +\infty$ , $ \mathbb E(Y^2) < +\infty $. How to prove that $|\mathbb E(XY)| \leq \sqrt{\mathbb V(X)\mathbb V(Y)}$ ? I tried to use ...
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0answers
50 views

Strict exogeneity assumption in OLS

Recently I read chapter 1 of Hayashi's Econometrics, the one explaining the finite-sample properties of OLS, and I had a doubt regarding the implications of the strict exogeneity assumption. Let's say ...
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0answers
26 views

Probability distribution of intersection of $0/1$ Bernoulli sequences

Assume odds of $1$ or $0$ is $0.5$ and independent. Suppose we have two randomly generated $0$ and $1$ sequences if length $2n$ each and each with number of $1$s between $0.5n - a$ and $0.5n + a$ ...
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1answer
55 views

What is the E[X| X, Y] when X and Y are dependent? [closed]

What is the $E[X| X, Y]$ when $X$ and $Y$ are dependent? Is it simply $X$? I was expecting this result because the knowledge about $X$ is already known (as we are conditioning on $X$ and $Y$). \...
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1answer
50 views

How to find the expectation of a Poisson process related variable

Q) Let $X_k\sim \operatorname{Exp}(\lambda)$, $X_k$ are iid which represent the interarrival times of a Poisson process of mean rate $\lambda$. Let $Y_k$ be the arrival times of the events. Let $$Z = \...
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1answer
63 views

When shouldn't you write $E(X)$?

We observe the discrete random variable $X = (X_1, . . . , X_n)$ with state space $S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family {${f_θ : θ ∈...
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0answers
27 views

How to derive $x$ w.r.t $p(\vert x\vert\le \vert a - f\vert\ , \vert x \vert\le \vert b - f\vert)$

How to derive the following probability in a close form formula with respect to $x$? \begin{equation} p(\vert x\vert\le \vert a - f\vert\ , \vert x \vert\le \vert b - f\vert) \end{equation} where $f \...
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1answer
34 views

Probabilities of the second highest roll in a dice pool.

I've been trying to figure out the probabilities of rolling a specific number in a pool of 4 20-sided dice, assuming I discard the highest and two lowest rolls. My instinct was to compound the ...
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0answers
28 views

What is the meaning of $\mathbb{E}^x$?

What does $\mathbb{E}^x((X_t)_{t \in T})$ for some stochastic process $(X_t)_{t \in T}$ mean? I heard that it is some kind of conditional expectation given that $X(0)=x$, but I can't find a precise ...
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0answers
42 views

Find $\lim_{n \to \infty} P(X_n+n = Y_n)$ If $X_n,Y_n$ have Poisson distribution of $Poi(n), Poi(2n)$

We are given a whole number $n \geq 1$ and independent Random variables $X_n$ with distribution $Poi(n)$ and $Y$ with distribution $Poi(2n)$. Determine $\lim_{n \to \infty} P(X_n+n = Y_n)$. I tried ...
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1answer
34 views

6-sides dice numbered from 1 to 6

A 6-sided dice has faces numbered 1,2,3,4,5 and 6 , when it is thrown the number facing up is the score , if it is thrown three times. a) Find the probability that the total score is 18. b) Find the ...
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3answers
376 views

A Game in Probability

You and n other players ( $n \ge 1$ ) play a game. Each player chooses a real number between 0 and 1. A referee also chooses a number between 0 and 1. The player who chooses the closest number to the ...
2
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1answer
49 views

Why does knowing X leads to a better guess on Y in terms of mean squared error?

Textbook : If we know $X=x$, then our best guess for $Y$ is $E(Y |X=x)$. Now $E(Y |X)$ is a random variable taking the value $E(Y | X = x)$ when $X = x$. So somebody who knows $X$ (whatever it turns ...
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1answer
31 views

Flip four coins. For every head, you get $\$1$. You may reflip at most two coins after the four flips. Calculate the expected returns.

I am modifying a question asked in MSE here. Flip four coins. For every head, you get $\$1$. You may reflip at most two coins after the four flips. Calculate the expected returns. In this case, ...
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1answer
49 views

Recommendations for probability text books? [duplicate]

I am trying to learn probability. Although I tried a few courses (Stat 110 Harvard for example), I felt the progression was fast and I kind of lost interest. However, I am dead-set on learning ...
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1answer
26 views

Help in defining the bounds of multiple integral

I am struggling with determining bounds of a quadruple integral and I hope that maybe someone might be able to help me out. I am looking into a situation where I have four independent, exponentially ...
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0answers
33 views

The Interpretation of Objective in Variational Autoencoder

The objective is, $\log P(X) - \mathcal{D}[Q(z|X)||P(z|X)]=E_{z\sim Q}[\log P(X|z)] - \mathcal{D}[Q(z|X)||P(z)]$. Obviously, $\log P(X) \geq E_{z\sim Q}[\log P(X|z)]$ (by Jensen's inequality). So we ...
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2answers
69 views

Stuck on End of Proving Bonferroni Inequality

An exercise asks me to prove "Bonferroni Inequality" and states it in the form of $$P\left( {A \cap B} \right) \ge 1 - P\left( {{A^C}} \right) - P\left( {{B^C}} \right)$$ I've searched a lot of old ...
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1answer
63 views

question about a problem from introduction to probability (bertsekas, 2nd, 2008)

Question: Your cousin has been playing the same video game from time immemorial. Assume that he wins each game with probability $p$. independent of the outcomes of other games. At midnight,...
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1answer
37 views

PDF of $X+Y$ and $X^2+Y^2$ when X, Y are independent uniform $[-1,1]$

Using convolution: $$ g_{X+Y}(t)=\int_\mathbb{R}g_X(t-x)g_Y(x)dx = \frac{1}{4}\int_{-1}^{1}\mathbb{1}_{[-1,1]}(t-x)dx = $$ Now, $t-x \in [-1, 1] \implies -x \in [-1-t,1-t] \implies x \in [t-1, t+1]$ ...
1
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1answer
26 views

What mean that Itô integral is not pathwise defined?

Let $(X_t)$ a stochastic process and denote $I(t)=\int_0^t X_sdB_s$. In a book I'm reading it's written : Not that we cannot say that at $\omega $, the integral depend only on the path $t\mapsto B_t(\...
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0answers
16 views

(equivalent) condition for a Markov chain to has a stationary distribution

Knowing about the existence of stationary distribution is an important question in Markov processes. Stationary distribution is the probability distribution $\pi$ with $\pi= \pi P.$ Let $(X_n: n \in \...
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0answers
31 views

Find the number of way in which the letters of the word 'EXTENSION' can be arranged in a straight line so that no two vowels are next to each other?

Find the number of way in which the letters of the word 'EXTENSION' can be arranged in a straight line so that no two vowels are next to each other? This is how i did it: _X_T_N_S_N_ '_' symbols ...
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0answers
27 views

Distribution of distances between two random lines passing through points at a given distance

I am trying to study the distribution of distances between 2 random lines in 3D passing each passing through one of the 2 points at a given distance $d$. For this, I am doing a 2 step process : 1. ...
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0answers
26 views

Integration a 2D Gaussian distribution over a line, what is the meaning of that and how to achieve that?

my problem is to find the best error model for a sensor. I got measurement(points) and reference surface(line segments) and I want a error model(2D gaussian distribustion) which gives the maximum ...
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1answer
40 views

How many 5-digit whole numbers with no 0's are divisible by 6?

How many 5-digit whole numbers with no 0s are divisible by 6? I've tried different methods to approach this question but still cannot get to the answer.
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0answers
61 views

Forecasting time series

I am trying to solve the following problem. Let the time series $S_t$, $t\in \{1,...N\}$ and consider its corresponding return $R_t$ defined as \begin{equation} R_t=\log(\frac{S_t}{S_{t-1}})=\mu_t+\...
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0answers
19 views

probability portion [closed]

In a certain population of hospital patients, the probability is 0.35 that a randomly selected patient will have heart disease. The probability is 0.86 that the patients with heart disease are smokers....
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0answers
30 views

Joint CDF of dependent variables

A random variable $\theta \sim U[ \underline{\Theta}, \overline{\Theta}]$ is drawn. Two signals are realised of the form $x_i=\theta +\sigma \epsilon_i$ when $\epsilon_i \sim U[-1,1]$ independent of $...
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1answer
30 views

Adding an explicit upper bound in the Strong Law of Large Numbers

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d variables with Bernoulli distribution $B(p)$. By the strong law of large numbers, we know that $\frac{X_1+\ldots+X_n}{n} \to p$ almost surely. My ...
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1answer
39 views

Expected value of couples: recurrence relation?

$n$ couples are formed randomly from an initial group of $n$ men and $n$ women. Denote men from $1$ to $n$ and let $X_i=1$ if the $i$-th man is associated to a women and $0$ otherwise. Let $X=\sum_{I=...
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2answers
84 views

Expected number of flips needed that at least two heads and one tail have been flipped

The original question is shown below (References from the book of Introduction to Probability Models Tenth Edition): A coin, having probability $p$ of landing heads, is continually flipped until at ...
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0answers
27 views

Computing probability of $k$ errors in a $q$-ary repetition code over a symmetric channel

I'm reading up on some coding theory and I'm having a hard time understanding error probabilities in $q$-ary repetition codes. Particularly, what I'm lost on conceptually is how to solve problems of ...
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0answers
17 views

Generalization of large deviation principle

My question arises from the following fact: For $X_i, i \geq 1$ i.i.d. standard normal distribution, we have for $\beta > 1/2$. Since $X_1 + \dots + X_n$ is a Gaussian random variable with mean $0$...
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1answer
30 views

Finding $c,b$ which minimizes $E(|X-c|)$ and $E[(X-b)^2]$

Let $X$ be a random variable with support ${1,2,3,5,15,25,50}$, each point of which has the same probability $\frac{1}{7}$. Argue that $c=5$ is the value that minimizes $h(c)=E(|X-c|)$. Compare $c$ ...
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1answer
32 views

Is it more probable to get a sum of 3 or 11 than it is of 2 or 12 when rolling two dice?

http://statweb.stanford.edu/~susan/courses/s60/split/node65.html Okay, so basically I was playing a board game and got into a debate on if a 3-11 is more likely than a 2-12. From my perspective I ...
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2answers
34 views

How to prove that $P(A \cap B) \leq P(A) + P(B)$ using $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

I need to prove that $P(A \cap B) \leq P(A) + P(B)$ using $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. So far I have: \begin{align*} P(A \cup B) & = P(A) + P(B) - P(A \cap B)\\ P(A \cup B) - P(A) -...
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0answers
61 views

Probability of employees absence

Given: On average one employee is absent 12 days per year (365 days). There are 5 employees in a company. It is assumed, that the absence of one employee has no effect on the others, and there is no ...
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0answers
16 views

Autocorrelation and The Markovian Assumption [closed]

I have a conceptual question that’d I’d love to hear back on, from any and all with knowledge of Markov chains/processes and time-series analysis. In my mind: the Markovian assumption implies that ...
1
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2answers
33 views

The probability of 1 green marble and no red marble being drawn

The question is A jar contains 3 red, 2 white, and 1 green marble. Two marbles are drawn at random without replacement from the jar. Let $X$ represent the number of red marbles drawn and let $Y$ ...
1
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1answer
34 views

Joint probability mass function of $\max(X,Y)$ and $\min(X,Y)-\max(X,Y)$ where $X,Y$ are independent geometric variables

Let $X$ and $Y$ be independent random variables with geometric probability function $p(k) = (1-\pi)\pi^{k}$ for $k = 0,1,...$ and $0 < \pi < 1$. Let $U = \max(X,Y)$ and $V = \min (X,Y)$. Define $...
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2answers
24 views

Panjer-Distribution and limit

Consider a discrete random variable $X \geq 0 $. Then there are $a,b \in \mathbb{R}: a+b>0 $, such that $$p_k = \left(a+ \frac{b}{k}\right) p_{k-1} $$ with $p_k=P(X=k)$ Now consider the case $a<...