# 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. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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### Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $R^2$. Calculate $Cov(X, Y)$

Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $\mathbb R^2$. Calculate $Cov(X,Y)$ I'm not sure how to solve this covariance problem. I would appreciate some ...
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### Probability density function of max(0, z)

Question The full question is from Exercise 4.6 in Statistical Inference (Casella, Berger). It is A and B agree to meet at certain place between 1 PM to 2 PM. Suppose they arrive at meeting place ...
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### PDF of following random variable.

I am trying to find the PDF of random variable $X$ but not getting it correctly. $X = \sum_{m=1}^{N}|a_m+\eta b_m c_m|^2$ ----(1) where $a, b, c$ are Zero mean circularly symmetric complex Gaussian (...
1 vote
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1 vote
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### Conditional distribution of one of the two exponential random variables, given one is smaller than the other

Let $X$ be a random variable with exponential distribution with parameter $a$, i.e. $X\sim Exp(a)$. See https://en.wikipedia.org/wiki/Exponential_distribution for the definition of exponential ...
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### Statistical mechanics partition function from probability distribution

I am curious about the mathematical background of something I came across while working on a problem in statistical mechanics. As an example, I am going to use the classical canonical ensemble, though ...
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### Estimates of parameters of interest $θ_1$ & $θ_2$ ($\pm$ standard errors) were $25\pm10$ & $10\pm3$. Find estimate & Standard Error of $δ=θ_1/5−θ_2$

Problem: In two independent studies, the estimates of the parameters of interest $θ_1$ and $θ_2$ ($\pm$ their standard errors) were computed to be $25\pm 10$ and $10\pm 3$, respectively. If one is ...
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### Given a person's age x, population's average life span u, predict life span.

Given a person's age x, population's average life span u, predict how long person can live? e.g. Say x is 6 and u = 76 one would expect life span of 74. But when x is 90, for same u = 76, one would ...
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### What is the distribution of a quadratic function of normal distributions? [closed]

Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$\sum_{i=1}^d (a_iZ_i+b_iZ_i^2).$$ I know when $a_i=0$, it is the ...
1 vote
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### Expected number of rounds for a product of uniform random variables on $[1/2,3/2]$ to be for the first time below a given threshold

Starting with w=1, each time we multiply w by a number x sampled independently and uniformly from [1/2, 3/2] until it is smaller than a given value c. What's the expected number of rounds for this ...
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### How a uniform distribution may induce different solution regarding Bertrand's paradox

I have a neophyte question regarding the formulation of the Bertrand's paradox. This is related to the definition of what we call a uniform distribution. If we consider the 3 point of views of “random ...
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### Data required for Dual Dirac PDF

I have a simple question regarding the dual Dirac PDF. If I have a set of deterministic data, e.g., d = [-2ps, 2ps, -2ps, 2ps, -2ps, 2ps] Would the resulting PDF look like a dual Dirac PDF? Where ...
1 vote
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### How the second inequality stands?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
1 vote
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### If a random variable has an atom at zero, does it have a density?

Let $Y$ be a random variable with distribution function $$F_Y(x) = \begin{cases} 0 &\quad x<0\\ p &\quad x=0\\ p + (1-p)F_X(x) &\quad x>0 \end{cases}$$ where $X$ is a continuous ...
1 vote
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### What is the result of XOR between two Bernoulli random variables?

Consider two Bernoulli random variables $X$ and $Y$ with probabilities $p_1$ and $p_2$. Now $Z = X \oplus Y$, $\oplus$ is a logical XOR operator. Is $Z$ a Bernoulli random variable and what is its ...
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### Probability of multiple distributions

I have a question concerning the calculation of the probability of an event among multiple probability distributions. Let's say we have two continuous probability distributions $D_1$ and $D_2$, given ...
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### Probability mass function of bank loan, find expected payment.

We have a random variable X that is the number of months that a certain owner of an estate needs to pay a loan in the Bank if he/she has a contract with insurance company to help him/her to pay a loan....