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

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0
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1answer
26 views

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 ...
0
votes
2answers
26 views

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 ...
1
vote
1answer
394 views

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) ...
0
votes
0answers
7 views

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 ...
2
votes
4answers
4k views

Addition of two Binomial Distribution

What is the distribution of the variable $X$ given $$X=Y+Z$$, where $Y$~Binomial($n$, $P_Y$) and $Z$~Binomial($n$, $P_Z$)? For the special case, when $P_Y = P_Z = P$, I think that X~Binomial($2n$, $...
3
votes
3answers
61 views

Flip $n$ coins, discard tails, and continue until $k$ heads remain

Consider the following game: $n$ participants have a fair coin each, on a given round, the not already discarded participants flip their coins, those who flip a tail are discarded from the game, the ...
1
vote
0answers
18 views

Normalization techniques for PDF

I have a function $$f:\mathbb{R}^n \rightarrow \mathbb{R}$$ which I would like to use as a probability density function. In order to do this I need to find a normalization constant $c$ so that when I ...
0
votes
0answers
17 views

Existence of Joint Distribution from Overlapping Marginal Distribution

Suppose $x_i\in \mathbb{R}^{n_i}$ for $i=0,1,...,k$. For each $i=1,...,k$, suppose $F_i$ is a probability measure of $(x_0,x_i)$ on $\mathbb{R}^{n_0 + n_i}$. Assume all $F_i$ have the same marginal ...
0
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0answers
13 views

Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $c$ such that $c\geq0$, and $\rho\geq0$. We want to prove ...
0
votes
2answers
38 views

Calculating the expectation of binomial distribution without calculating the summation

We know that expectation of a binomial distribution is $$\sum _{1}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{n-k}k = np$$ But while proving it, it is being written ...
1
vote
1answer
20 views

Joint probability density function $(X^2,Y^2)$

Let $X$ and $Y$ be ramdom variables having the following joint probability density function $f(x,y)=\begin{cases} \frac{3}{8}xy & x\geq0,\,y\geq0,\:x+y\leq2\\ 0 & otherwise \end{cases}$ ...
1
vote
1answer
37 views

Scale invariance of uniform distribution over $\mathbb R^2$?

If we make a uniform distribution of points over $\mathbb R^2$ with 1 point on average per unit square. And we zoom far out and make a density plot (give a color to each cell according to how many ...
0
votes
2answers
29 views

Value of c so that $c(2-|x|-|y|)$ is a probability distribution function(see picture)

Hint: Use the formula of volume of pyaramid. My approach: I know that the integral of a pdf from $-\infty to +\infty$ gives you $1$. I tried taking the double integral, but got stuck in as how to ...
1
vote
1answer
682 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
0
votes
1answer
375 views

Finding joint pmfs from marginal pmfs

Let a, b > 0. The random variables X and Y are independent and their densities are : f(x) = 1/gamma(a)*x^(x-1)*e^-x, x>= 0 f(y) = 1/gamma(b)*y^(b-1)*e^-y, y>= 0 Let U=X+Y and V=X/X+Y Find the ...
2
votes
0answers
32 views

PDF/CDF of max-min type random variable

For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$ When we have $X=\...
0
votes
1answer
28 views
+50

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 ...
1
vote
1answer
22 views

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 ...
3
votes
0answers
25 views

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 ...
0
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0answers
12 views

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 $x$ ...
9
votes
4answers
19k views

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 ...
2
votes
1answer
29 views

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 ...
0
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0answers
18 views

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 ...
0
votes
1answer
20 views

Is this equality holds? $\overline{F^{*2}}(x)=\int_0^x\overline{F}(x-y)dF(y)$

$X_1,X_2$ are non-negative i.i.d random variables with CDF F(x). I have a problem proving that following identity holds. $$ \frac{\overline{F^{*2}}(x)}{\overline{F}(x)}=1+\int_0^x\frac{\overline{F}(...
1
vote
1answer
789 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 \...
0
votes
1answer
14 views

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 ...
1
vote
4answers
2k views

Expectation Poisson Distribution

A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and $10,000 for each one ...
0
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0answers
17 views

Fuzzy variant of cos²(x)

For a simulation, I need to find a variant fuzzy function of cos²(x) For the intented purpose, with n from 5 to 20 and L small , the function $fuzzyCos(a) = \prod_{i=1}^{n}{ cos^2( a + i \frac{L}{n} -...
0
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0answers
12 views

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 ...
1
vote
2answers
25 views

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|...
0
votes
0answers
25 views

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 ...
2
votes
0answers
62 views

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 ...
0
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0answers
18 views

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 ...
1
vote
0answers
59 views

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 ...
2
votes
1answer
37 views

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 ...
1
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0answers
24 views

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 ...
-3
votes
1answer
41 views

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?
1
vote
1answer
36 views

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 ...
-3
votes
0answers
23 views

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
-4
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3answers
62 views

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: $$\int_{-2}^2\int_{-\infty}^x f(u)\,du\,dx=\cdots\text{?}$$
2
votes
1answer
40 views

Showing that if $X \sim \operatorname{Exp}(1)$, then $Y = F_X(X)$ has uniform distribution on $[0,1]$

Let $X \sim \operatorname{Exp}(1)$, and show $Y = F_X(X)$ has uniform distribution on $[0,1]$. I calculated $F_Y$, since the cumulative distribution function identifies a distribution. We have: $$\...
1
vote
0answers
18 views

Multivariate normal distribution conditional probability question.

$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$We have that $X$ and $Y$ are random variables with a ...
0
votes
0answers
29 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
1
vote
1answer
397 views

How do i calculate the probability of the relay in the circuits?

I am trying to solve my following probability question but i can't see how to make any progress. Any help will be highly appreciated Question: The probability of the closing of the i-th relay in the ...
1
vote
1answer
55 views

Expectation of $|H - T|$

Using binomial approximation to normal distribution, find the expectation of $|H-T|$ where the $H,T$ are heads and tails of a fair coin and the number of tosses is large. Can anyone please tell me, ...
0
votes
0answers
17 views

How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
4
votes
1answer
1k views

Inverse Mills ratio for non normal distributions.

We have the well known result of the inverse Mills ratio: $$ \mathbb{E}[\,X\,|_{\ X > k} \,] = \mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
1
vote
1answer
43 views

Inequality: product of integrals

Context: Proving integral inequalities about posterior distributions following different sequences of binary signals. The proofs come down to the following inequalities. Let $\psi(x)$ be a concave ...
0
votes
1answer
52 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...
0
votes
3answers
30 views

How to find E(x) and Var(x) in this specific continuous probability distribution.

I've got into some confusion on continuous probability distributions, and everything related to it. This is the problem: Problem Image. I assume from the sketch that pdf is $f(x) = x$ for values of $x$...