# 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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### we use the poisson random distribution

Let ܺ denote the number of typographical errors on a single page of the lecture notes. Let’s assume that ܺ has Poisson distribution with parameter ߣ = 1. Calculate the probability that there is at ...
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### how can we derive both the CDF of $M$ and the PDF of $M$

Given $X_i \sim Uniform[0, 1]$ for $i = 1, \dots, n$. What is the distribution of $M := \min(X_1, \dots, X_n)$? I feel like I'm missing something and I've been stuck on this for two days
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### Probability of difference between any two random variables

Given random variables $x_1,x_2,\ldots,x_n$ which are i.i.d. with $x_i \sim \mathcal N(\eta, \sigma^2), \forall 1 \leq i \leq n$. I tried to find the probability that none of these random variables ...
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### A general expression for the sum of multiple independent Normal Mixture Distributions?

Suppose random variable $X_1$ is a mixture of two Normal distributions with means of $\mu_A$ and $\mu_B$ respectively, standard deviations of $\sigma_A$ and $\sigma_B$ respectively, and weights given ...
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### Relating Total Variation Distance to Number of Samples

I have a dataset whose underlying joint distribution is $P_{XY}$. $X$ is the input and $Y$ is the output. I restrict someone to take $n$ samples from this dataset. Based on the samples he got, he ...
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### Determining the distribution and showing it's a pivot

so we let $X_1....X_n$ be iid random variables from the exp($\theta$) distribution. Find the distribution of S=$2\theta\sum_{i=1}^n(X_i)$ and hence show that S is a pivot. Now i understand that the ...
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### $X\sim\text{Unif}[-1,2]$ find PDF of $Y=X^2$

$X\sim\text{Unif}[-1,2]$ find PDF of $Y=X^2$ Would someone mind explaining to me the parts of the solution I'm confused about? "Since $-1\le X\le2$ and we have $0\le X^2\le4$" I understand the 4 ...
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### Expected value of truncated distribution

This definition is taken from the Wikipedia page for truncated distribution: Let $X$ be a random variable with a continuous distribution, $f(x)$ be its probability density function and $F(x)$ be its ...
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### Does exist other techniques for finding PDF/CDF from relation between random variables?

I know two techniques of finding CDF/PDF from relation between random variables. By relation I mean that one rv is represented by other, examples $Y = X^2$ or $Z = \frac {U} {1-U}$. Summarizing, ...
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### How to see a function is Gaussian

I have this function $$f(x, y) = \frac {1}{2\pi}\exp(−0.5(x^2-2xy+9y^2))$$ I proceed like this: First I compute $\Sigma^{-1}$ which is \begin{bmatrix} 1 & -1 \\ -1& 9\end{bmatrix} Then I ...
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### Covariance Formula

Hy everybody, I have $f_X(x)$=$(3/14)(x^2+2)$ and $f_Y(y)$=$3/28((1/3)+y)$ and the joint $f(xy)$=$(3/56)(x^2+y)$ I have to compute the $Cov(XY)$, I proceed like this: I know they are dependent, ...
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### Derive the t-distribution using transformation of random variables.

I'm asked to derive the pdf of the t-distribution following way. $$V = \frac{Z}{\sqrt{U/n}}$$ where $Z$ is the standard normal and $U$ is the chi-squared distribution with degree of freedom n. ...
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### Topic-prevalence in Latent Dirichlet Allocation

In Focused Topic Models, the main motivation is to decouple the global topic prevalence and in-document prevalence. However, I couldn't see how the original Latent Dirichlet Allocation (LDA) couples ...
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### Coin flipping game - expected value

Given a weighted coin that lands on heads 55% of the time, you flip the coin until you get you get your first tails. For each heads, you make \$1. When you flip tails you lose \$0.90. What is the ...
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### $A, B, C$ $IID$ $geometric(p);$ $E[\frac{A - 2B + C}{A}]$ [duplicate]

I know how to do all the steps but I am stuck when calculating $E[\frac{1}{A}]$. Any properties of the geometric distribution (i.e apart from solving using the pmf) to solve this? And for people who ...
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### conditional expectation of uniform distribution given the realization of product of uniform distributions

Assume that $A \sim U(0,1)$, and $B \sim U(0,b)$ with $b<1$, and A and B are independent. Can we calculate (closed form for) the expected value of $A$ given that we observe the realization of AB, ...
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### Sequences of probability distributions which do not converege uniformly and satisfy integral properites.

I'm looking for two (convergent) sequences of real-valued functions, $\{f_{k}\}$ and $\{g_{k}\}$, such that, for each $k$, $$\int_{-\infty}^{x}f_{k}(t)dt\leq \int_{-\infty}^{x}g_{k}(t)dt$$ at every $x$...
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### In a MLE what are the rules for getting things out of the product symbol $\prod$ in our $f(x_1,…,x_n | \theta )$?

In a MLE what are the rules for getting things out of the product symbol in our $f(x_1,...,x_n | \theta )$? What I mean is once we have the density function of our distribution we write \$f(x_1,...,...