# Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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### Is the CMF of a log-concave PMF also log-concave?

If a PDF is log-concave, then its CDF is also log-concave. The proof I know for this uses the derivative of the log function, see Proposition 1 in this paper. Does this also hold for discrete ...
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### Compare two coin tossing games

Compare the following two games: You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game ...
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### Sum of independent exponential distributions?

A person has $100$ light bulbs whose lifetimes are independent exponentials with mean $5$ hours. The bulbs are used one at a time, with a failed bulb being replaced immediately by a new one. (a) ...
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### Computationally Efficient Way to Partition N-Dimensional Space Around Distinct Values

Sorry if the title isn't super helpful, I'm really just looking for someone to point me in the right direction or let me know if there is a standard way of doing this. What I am wondering is, if I ...
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### Transforming a categorical distribution by repeating trials and taking a plurality

Suppose you have a K-sided, weighted die. This is represented by a categorical distribution. Now, let's say you roll the die N times, and then pick a "winner" by choosing whichever outcome has a ...
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### Basic query Related to dependent random variables

$X$ and $Y$ are two dependent random variables. I want to find the following probability $$\Pr(2X<c,4Y>c)$$ wher $c$ is some positive number. In my attempt, I can expand the above probability ...
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### Distinct items in a Sample of a Zipfian Distribution

The Zipfian distribution serves as a good model for several interesting things. For example, the rate of occurrence of words in the English language (or most any language) appear to follow a Zipfian ...
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### Help needed related to derivation

I want to find the following probability $$P(z_i\leq min(1,x^{-m})<z_{i+1}, x<x_1|z_i \leq 1, z_{i+1}>1)$$ where $m$ is some value greater than $2$, $z_i$'s are some constants and pdf of $x$ ...
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### Difference between power law distribution and exponential decay

This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like ...
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### Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
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### Difference between a mixture of distributions and a convolution. Intuition

From what I could gather Mixture: if $X_i\sim^{iid} f_i$, then W is a mixture with $f_W =\sum \frac{f_i}{n}$. This definition could also be for the CDF instead of the density. Convolution: To make ...
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### Find the limit of the following series of normal random variables.

Let $X_1,X_2,X_3,…$ be a sequence of i.i.d. $N(\mu,1)$ random variables. Then, find $$\lim_{n\to \infty} \frac{\sqrt{\pi}}{2n}\sum_{i=1}^{n}E(|X_i-\mu|).$$ My thoughts: I don't have any rigorous way ...
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### Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
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### Calculate probabilies based on given probability distribution

A mail-order company business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table \begin{array}{r|...
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### distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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### When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
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### Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
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### Big Balloon Game

The problem In this game, you are given empty balloons one by one, and for each balloon you are to inflate it with air until you are satisfied. If it does not burst, you gain happiness points ...
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### determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
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### Is there a known probability distribution with cumulative probability function $F(X)=\frac{1-X^a T_1}{1-X^a T_2}$? [on hold]

I have a random variable $X$, which is distributed with the cumulative probability function $F(X)=\frac{1-X^a T_1}{1-X^a T_2}$, where $a$ is negative. I am wondering is there any famous distributions ...
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### what is the approximate probability that you win more than 120 times if you purchase 900 tickets? [closed]

The fine print on an instant lottery ticket claims that one in nine tickets win a prize. What is the approximate probability that you win more than 120 times if you purchase 900 tickets?
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### Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused ...
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### I am not clear about the limits of x and y. [closed]

Let X and Y be continuous random variables with joint probability density function f(x,y)=exp(x+y);- infinity
### i want to know that how to find $f(u)\,du$ here??? [closed]
The probability density function $f(x)$ of a random variable $X$ is symmetric about zero. Then: $$\int_{-2}^2\int_{-\infty}^x f(u)\,du\,dx=\cdots\text{?}$$