# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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### Distribution of the sum of absolutes values of T-distributed random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed independent variables $\sum_{1 \leq i \leq n}|x_i|$....
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### Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
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### Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
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### Enhanced Berry-Esseen theorem for the digits of $\sqrt{2}$

The Berry-Esseen theorem provides a second-order approximation to the central limit theorem (itself a first order approximation.) Higher order approximations are available, see here. If the random ...
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### Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
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### What is the Lévy measure of the Student's $t$-distribution?

It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a ...
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### Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R$ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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### Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
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### What is the distribution of the difference of two normalized binomial random variables?

Let $X \sim Bin(n, p)$ and $Y \sim Bin(m, p)$. How is $$Z_1 = \frac{X}{n} - \frac{Y}{m}$$ and $$Z_2 = \left|\frac{X}{n} - \frac{Y}{m}\right|$$ distributed? (Hence: What is their cumulative ...
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### Definition of expectation value in quantum mechanics

I've read the following proposition in a book on quantum theory. Proposition. If a quantum system is in a state described by a unit vector $\psi$ and for some quantum observable $\hat{f}$ we have ...
### PDF of difference of two i.i.d. random variables: maximum at $0$ and decreasing to the right of $0$?
Let $X, Y$ be two i.i.d. random variables with an arbitrary distribution. As discussed here, the distribution of their difference $X-Y$ is symmetric around $0$. What I am wondering: Does this ...