# Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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### Are all random variables estimators? [duplicate]

My hand-wavey understanding is a random variable is a function from a domain of possible outcomes in a sample space to a measurable space valued in real numbers. We might denote a random variable from ...
21 views

### Proof of Strong consistency of Beta posterior distribution

Suppose that we have random variable $X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0})$ with $p_{0}$ true unknown probability in $[0,1]$. Now, I want to implement Bayesian machinery to ...
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### Probability of fixing error in application, given successful and unsuccessful runs before fix

An application communicating with a database started to fail with network errors. The errors occur only during some runs, and at random times when running the application. If a network error occurs, ...
• 201
2k views

### Equivalence of first/second choice with naive probability - I don't buy it

I'm seeking a better understanding of the following problem from Blitzstein and Huang (2015) (Chapter 1, Exercise 31, p. 35): A jar contains $r$ red balls and $g$ green balls, where $r$ and $g$ are ...
• 161
1 vote
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### Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? [closed]

Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances? Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
• 139
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### Bayes' Theorem applied in real study [closed]

My name is Molly, and I am a medical student at Queens. I'm delving into the realm of statistics. It's essential to comprehend how statistical methods are utilized in medical screening and diagnostics....
129 views

### Equivalence of inverse transformations under distributional equivalence

Consider continuous, invertible transformations $g,h : \Bbb{R}^d \rightarrow \Bbb{R}^d$ and suppose $g(Y) \overset{d}{=} h(Y)$, where $Y$ is a $N(0, I)$ random variable. Then what can we infer about ...
• 51
1 vote
35 views

### relating correlation to probabilities

Given two centered and scaled random variables $X$ and $Y$, can you relate the probability they have the same sign to their correlation? If the correlation is close to $1$, I am picturing the joint ...
• 149
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### Let $X(t)$ be a Gaussian process. Does $\mathbb{E}[X(t)^2 X(s)^2] = \mathbb{E}[X(t)^2 ] \mathbb{E}[X(s)^2 ] + 2 (\mathbb{E}[X(t) X(s)])^2$?

As the title says, can I apply Isserlis' theorem to $\mathbb{E}[X(t)X(t)X(s)X(s)]$?
44 views

### 100 prisoners problem with n < 50

In the 100 prisoners problem, I understand that when $l \geq 51$, the number of permutations is $100!/l$. However how to calculate the exact number of permutations when $l < 51$? When $l = 1$, the ...
• 203
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### Bayesian network with partial info (4 nodes)

I have some (conditional) probabilities for a Bayesian network with binary variables, but not all. My DAG is M->F->Y->C<-F and M->Y and M->C I ...
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1 vote
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### Markov Chains with Changing Number of States

I have seen these kinds of Discrete State Markov Chains before (Continuous Time or Discrete Time): Homogeneous (Probability Transition Matrix is constant) Non-Homogeneous (Probability Transition ...
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### Data generation processes that "vary slowly enough"

I am considering some abstract data generation processes producing data by first drawing features $x_{1:n}\sim p(x)$ iid, and then drawing the responses $y_i\sim p(y\mid x_i)$ iid, where $p(y\mid x)$ ...