Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
2
votes
1answer
57 views
Confusion about Banach Matchbox problem
While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand.
The problem statement is presented below (Source:Here)
...
1
vote
1answer
35 views
Question on MGFs & Finding The Distribution of a Sum
Suppose that $X_1$, $X_2$, ..., $X_n$ are independent, where each $X_i$ has probability (mass) function $p_i$($x_i$) given as follows:
$p_i$($x_i$) = $\frac{e^{-\lambda}\lambda_i^{x_i}}{x_i!}$ (the ...
1
vote
1answer
34 views
How many times must I repeat a trial to have confidence in a true result?
How can I calculate the number of times I need to repeat an independent test with a probability (p) in order to have 99% confidence that I will have at least 1 true result?
Example:
I have a task ...
0
votes
1answer
35 views
HELP Distribution of the Minimum of two random variables
Well Let $Y$ be a random variable that could be discrete or continuous and $M$ a positive constant random variable Find the distribution of $S$$=$$min${Y,M}
My progress so far is :
$p($S ...
1
vote
3answers
67 views
Apparent contradiction for probability density functions?
Consider a probability density function $\it{pdf}$, $f\left(x\right)$, which can be expanded as:
$$ f\left(x\right) = \sum_{k=1}^{\infty} \alpha_k \delta\left(x-x_k\right)$$
It is easy to verify, by ...
0
votes
1answer
24 views
Approximating Chi squared distribution
A machine in a heavy equipment-factory produces steel rods of length Y , where Y
is a normally distributed random variable with mean 6cm inches and variance $\frac{1}{4} cm^2 $. Thecost C of repairing ...
1
vote
1answer
42 views
How to plot multinomial beta from Dirichlet Distribution
I would like to plot the multinomial beta from the Dirichlet Distribution wiki page.
$$\mathrm{B}(\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K ...
0
votes
2answers
55 views
Probability distribution of a sum of uniform random variables
Given
$$X = \sum_i^n x_i$$
,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.
1
vote
0answers
23 views
What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]
Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads?
How do I apply Chernoff Bounds to this?
I really need help understanding Chernoff Bounds.
0
votes
1answer
12 views
Right-skewd or power law distribution
Is a power law distribution right skewed?
Is there any relation between concepts of right skewness and power law distribution?
1
vote
1answer
56 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
1
vote
0answers
20 views
A question about monotonicity
Is
$$D(y_l)=\int_{-\infty}^{y_l}f_0(y)\mbox{d}y+\int_{y_l}^{y_u}e^{x\ln(1/L(y_l))}L(y)^{x}f_0(y)\mbox{d}y+\frac{1}{L(y_l)}\int_{y_u}^{\infty}f_0(y)\mbox{d}y$$
with
...
5
votes
0answers
64 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
2
votes
1answer
46 views
Using empirical density function as an estimator of a given probability density
We know empirical distribution function is defined as $F_n(x)=\frac{1}{n}\sum\limits_{i=1}^nI(X_i \leq x)$. Then define empirical density function as $ f_n(x) = \frac{F_n(x+b_n)-F_n(x-b_n)}{2b_n} $ .
...
3
votes
1answer
38 views
poisson distribution jobs in printer
A printer receives a number of jobs in an hour, which is poisson distributed with parameter $\lambda$. Every job is recognized with a probability $p$ such that the job is faulty and wont be printed.
...
0
votes
1answer
45 views
Conditional Density, Additive Gaussian
A signal, X, is a random variable with the following density function:
$$f_X(x) =\begin{cases} \frac{3}{25}(x-5)^2, &0 \le x \le 5\\0, &otherwise \end{cases}
$$
The signal is transmitted ...
0
votes
0answers
16 views
compare multi-distribution
If
$Y$= a random variable, it is a product of 2 independent uniform distributions [0,1]
$X_1$ = a random variable, it is a independent uniform distributions [0,1]
$X_2$ = a random variable, it is ...
1
vote
1answer
31 views
Evaluate the expectation of $\frac{1}{UV}$
Another question from the past test papers!
The joint density function of $X$ and $Y$ is given by $f_{X,Y}(x,y)
= \frac{1}{x^2 y^2}$, $x \geq 1, y \geq 1$. (i) Find the joint density function of ...
1
vote
1answer
21 views
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
I am trying to derive Chi-square distribution. The random variale is
$$ U^2=\sum_{i=1}^k X_i^2 $$
where $X$ is a random variable with normal standard distribution.
What is the distribution of ...
1
vote
0answers
59 views
Probability distribution with maximal entropy on $[0,1] \cup \{2\}$
For given closed set $F$ on $\mathbb R$ one can think of probability distribution $\mathbb P^\ast_F$ with support on $F$ and with maximal entropy. It is well known that
If $F=[0,1]$ then $\mathbb ...
0
votes
1answer
28 views
Probability Distribution Of A Linear Combination
I have scoured my textbook for the concept alluded to in the title of this thread; however, my textbook has failed, in that it provides no such information.
Does anyone know of some resources for ...
1
vote
1answer
34 views
Looking for a probability distribution in $\mathbb{R}^n$: exponentially decreasing density as distance from point
I was thinking of a probability distribution over points in $\mathbb{R}^n$ of the following form. Let $\mu \in \mathbb{R}^n$, $b$ a scale parameter, and $X$ a random variable in $\mathbb{R}^n$; then ...
1
vote
0answers
21 views
Discrete probability and approximation to hoeffding inequality
A sample of heads and tails is created by tossing a coin a number of times independently. Assume we have a number of coins that generate different samples independently. For a given coin, let the ...
2
votes
1answer
46 views
A Problem on Uniform Probability Distribution
Consider three independent uniformly distributed (taking values between 0 and 1) random variables. What is the probability that the middle of the three values (between the lowest and the highest ...
-1
votes
6answers
73 views
Density function
Let $f(x) = e^{{-x -e}^{-x}} $ .
How can I check that f is a density function?
I know that it has to be valid that $ \int_{-\infty}^{\infty}{f(x)} = 1 $ , but how to check this?
0
votes
1answer
21 views
Coding Distributions as a Convex Constraint
In convex optimization, how can we impose a constraint that a variable has certain distribution?
e.g. elements of vector $v$ have power law distribution?
1
vote
2answers
55 views
A problem on continuous i.i.d random variable
Let $X_1, X_2, X_3, X_4$ be i.i.d continuous random variables with a common distribution function $F$. How to prove that "all the 4! possible orderings of $X_1, \dots, X_4$ are equally likely" without ...
3
votes
1answer
37 views
Why is this Poisson distribution incorrect?
Assume power failures occur independently of each other at a uniform rate through the months of the year, with little chance of $2$ or more occurring simultaneously. Suppose that $80\%$ of months have ...
0
votes
0answers
18 views
What convex combinations of Chi Squared i.i.d random variables minimizes/maximizes their tail distribution
Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& ...
0
votes
1answer
30 views
Determine the Distribution Function
Another density function to Distribution function that has me stumped again. This time, I might be going wrong on my intervals.
$$
f(x)=
\begin{cases}
x+1, & -1<x\le0\\
1-x, & 0< ...
2
votes
1answer
62 views
Is this linear programming?
I have the following problem and I'd like to know if it is formalizable as a LP program.
(or, more generally, if it is solvable in polynomial time).
Let us fix some terminology first which will ...
1
vote
1answer
45 views
maximum of exponentials
I am really having difficulties to prove the following: consider $X_1,\dots, X_n$ all exponentially distributed with rate $\lambda$ (i.e. $X_i \sim exp( 1/\lambda)$). Then argue that we can write
...
2
votes
1answer
50 views
Limit of Binomial distribution
In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
0
votes
1answer
24 views
Value of c in a density function
I've been trying to figure out how the answer was obtained with this integral and I'm completely stuck. Please help explain how was this integrated to get to the answer...
For positive integer n, ...
0
votes
0answers
38 views
Find the distribution of min(U,V)/max(U,V) for independent uniform (0,1) variables U and V.
Find the distribution of "min(U,V)/max(U,V)" for independent uniform (0,1) variables U and V.
0
votes
0answers
49 views
something about property of Bernoulli random variables
Let $b_i, i=1, \ldots, n$ be Bernoulli random variables with probability $P(b_i=1)=2k/n,$ where $k\leq n.$
Show the following:
Let $\chi$ be an indicator function that $k$ out of $n$ of $b_i$ are ...
0
votes
1answer
56 views
Conditional expectation $E[X|Y<y]$
Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.
Is it true that $E[X|Y < y]$ equals:
$$ \displaystyle ...
1
vote
1answer
39 views
Convergence in distribution - Gamma distribution
If we have a random variable defined as $Y_{n}=\displaystyle\frac{X_n-n\alpha}{n\alpha^2}$, where $X_n$ is $\operatorname{Gamma}(n,\alpha)$ distribution, how can I prove that $Y_n$ converges in ...
0
votes
0answers
34 views
Can Bhattacharyya distance be greater than one?
I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ ...
0
votes
0answers
25 views
Probability Density Function and Eigenvalue Spectrum of Correlation Matrix
My question is in the link...
http://www.flickr.com/photos/88684900@N03/8654322505/in/photostream
0
votes
0answers
28 views
What is a topic I could easily collect data on that follows a poisson distribution?
So I have a project for my stat class and we have to form a hypothesis or question that I can do probabilistic modeling on. I really want to do a topic similar to a poisson distribution but I'm a ...
0
votes
0answers
23 views
Finding the joint distribution of 2 ratio of Gaussian random variables
Given independent normal random variables $X$, $Y$, and $Z$, I have the following ratios defined
$$
\begin{align}
r_1 &= \frac{x}{z}
\\
r_2 &= \frac{y}{z}
\end{align}
$$
The marginal ...
1
vote
0answers
38 views
What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?
It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
0
votes
1answer
61 views
Cumulative distribution function or density for Compound Poisson distribution
I have the Compound Poisson distribution
$$ \xi = \sum_{n=1}^N X_n $$
where N has Poisson ($\lambda$) distribution and $X_i$ are independent and identically distributed and have normal distribution. ...
2
votes
0answers
37 views
Useful approximation of the pdf
Good day to everyone.
In my research work I came out with a function, which looks like this (it is the pdf of some random variable):
$$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
1
vote
1answer
21 views
Stochastic variable equals indicator function?
An exercise in my statistics & probability theory course goes as follows:
$\Omega = [0,1], \mathcal{B} = \mathcal{B}([0,1]), P$ the Lebesgue measure on $[0,1]$.
We have the sequence of ...
1
vote
1answer
50 views
Statistics and Probabilities- Distributions
A quality control engineer tests the quality of produced computers. Suppose that 5% of computers have defects and defects occur independently of each other. I need to find the probability that the ...
0
votes
0answers
10 views
Multiplication of Multivariate T distributions
I need to integrate two (unnormalised) multivariate T-distributions $\int t_{\nu_1}(x|0,C_1)t_{\nu_2}(x|\mu,C_2) dx $. Note that they are of two different degrees of freedom which makes the question ...
1
vote
1answer
38 views
Are the multiplications of i.i.d random variables , i.i.d?
If we know that $X_1$ and $X_2$ are i.i.d random variables, and $Z_1$ and $Z_2$ are also i.i.d random variables, can we say $X_1Z_1$ and $X_2Z_2$ are i.i.d random variables too? suppose that $X_1$ ...
1
vote
0answers
12 views
Suggest Power law distributions reference
What is everyone suggestion for a good Power Law Distribution reference in book or article form? Both theory and application is helpful (two separate books is fine).
Thank you.
