Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
7
votes
2answers
1k views
Given the pdf of independent RVs $I$ and $R$, how to find cdf of $W =I^2R$?
Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$?
Where,
$$
\begin{align}
f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\
f_R(r)&=2r, &0 \leq ...
5
votes
2answers
2k views
Proof of upper-tail inequality for standard normal distribution
$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$,
$$
\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.
$$
5
votes
3answers
765 views
Proof that the sum of two Gaussian variables is another Gaussian
The sum of two Gaussian variables is another Gaussian.
It seems natural, but I could not find a proof using Google.
What's a short way to prove this?
Thanks!
Edit: Provided the two variables are ...
9
votes
3answers
2k views
Expectation of the maximum of IID geometric random variables
Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is
$$E_n = E\left(\max_{i \in 1 .. ...
3
votes
2answers
2k views
Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$
It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
2
votes
1answer
199 views
probability distribution of coverage of a set after `X` independently, randomly selected members of the set
I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
1
vote
1answer
505 views
Multivariate normal and multivariate Bernoulli
Say I only have the mean vector and the covariance matrix of some multivaraite distribution X, where all single-variable marinals are normal (note: this is not generally a multinormal distribution). ...
6
votes
1answer
369 views
Limit using Poisson distribution
Show using the Poisson distribution that
$$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
5
votes
2answers
240 views
Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below:
$\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$
Can it be proven ...
5
votes
3answers
472 views
Quantile function properties
I am confused by "Inverse distribution function (quantile function)" section of the wikipedia page on CDFs
. It says that $$F^{-1}(F(x)) \leq x\text{ and }F(F^{-1}(y)) \geq y$$
However, I ...
5
votes
1answer
99 views
Help with convergence in distribution
$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$
Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity?
I let $Z = ...
2
votes
2answers
335 views
Computation of the probability density function for $(X,Y) = \sqrt{2 R} ( \cos(\theta), \sin(\theta))$
Let $R$ be a almost surely non-negative continuous random variable with absolutely continuous measure, and $\Theta$ be an independent random variable, uniformly distributed on the interval $[0, 2 ...
2
votes
1answer
577 views
Probability of $n$ successes in a row at the $k$-th Bernoulli trial… geometric?
If one has Bernoulli trials with success probability $p$, then it makes sense that the probability of the first success observed to be at trial number $k$ be given by
$$(1-p)^{k-1} p.$$
But how ...
5
votes
3answers
342 views
How to calculate $E[(\int_0^t{W_sds})^n], n \geq 2$
Let $W_t$ be a standard one dimension Brownian Motion with $W_0=0$ and $X_t=\int_0^t{W_sds}$.
With the help of ito formula, we could get
$$E[(X_t)^2]=\frac{1}{3}t^3$$
$$E[(X_t)^3]=0$$
When I try to ...
10
votes
3answers
251 views
Random point uniform on a sphere
If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$, Are the coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$?
5
votes
2answers
497 views
Uniform distribution with probability density function. Find the value of $k$.
For a random sample $X_1,X_2,...X_n$ from a uniform $[0,\Theta]$ distribution, with probability density function
$$f(x;\Theta) = \left\{ \begin{array} \ \frac{1}{\Theta} & 0\le x \le\Theta,\\ 0 ...
5
votes
1answer
814 views
Distribution of Ratio of Exponential and Gamma random variable
A recent question asked about the distribution of the ratio of two random variables, and the answer accepted there was a reference to Wikipedia which (in simplified and restated form) claims that if ...
5
votes
2answers
502 views
Showing $\cos(t^2)$ is not a Characteristic Function
Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
3
votes
2answers
1k views
Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT
Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
-3
votes
1answer
736 views
Calculation of a most probable value based on multiple probability distributions
Given are multiple probability distributions for a random variable.
Is there a way to calculate the most probable value based on these distributions?
Example:
...
6
votes
1answer
557 views
Distribution of the digits of Pi
Can anything be stated about the distribution of the digits of Pi, i.e., if I were to sample n digits of Pi, can anything be said about the probability to observe certain digits, or is there any ...
4
votes
2answers
53 views
finding the number of circles we get when randomly placing given patterns into a grid of squares
We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into ...
3
votes
1answer
96 views
Truchet tiles on a flattened cube
We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
3
votes
1answer
240 views
Question about order statistics
I saw a paper which says that:
Let $Z_i$ be i.i.d. exponential random variables with mean $1$, and let
$S_n = Z_1 + \dots + Z_n$ for all $n$. For a fixed $n$, let $U_j = S_j/S_{n+1}$, then ...
2
votes
2answers
107 views
Other way to express $e^{|x|+|y|}$
I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
votes
2answers
89 views
Distribution of sums
I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that ...
2
votes
4answers
720 views
Derive the expected value for a Pareto distribution?
X is a random value that is Pareto distributed with parameter $a>0$, if $\Pr(X>x)=x^{-a}$ for all $x≥1$.
Show that $EX=a/(a-1)$ if $a>1$ and $E(X)=∞$ if $0< a \le1$.
I can derive the ...
2
votes
1answer
270 views
Probability distribution of sign changes in Brownian motion
Let us consider a 1d Brownian motion. Displacements in space will be positive or negative and this is a random variable $U(t)$ that characterizes a random process and that can take just the values ...
2
votes
1answer
280 views
Exponential distribution from Poisson
In Poisson distribution, the probability of inter arrival time to be t or less is:
$$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$
and probability of one ...
1
vote
1answer
438 views
Coin toss - probability of a tail known that one is heads
A friend of mine tossed a fair coin twice. Suppose I ask him whether he got a head in the two tosses, and he says yes. What is the probability that one toss is tail?
Now suppose instead that I happen ...
1
vote
3answers
822 views
Poisson Distribution of sum of two random independent variables $X$, $Y$
$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim ...
1
vote
1answer
229 views
probability question on characteristic function
I got a big problem with my exam practice question on characteristic function. Here are two.
Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
1
vote
2answers
123 views
A question about sampling distribution
Assume there are $n$ independent random variables $X_1,X_2,\ldots,X_n$ and i wonder why the sample variance is $S^2=\frac{\sum\limits_{i=1}^n \ (X_i-X)^2}{n-1}$ where ...
1
vote
1answer
326 views
Prove that vector has normal distribution
You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$.
Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$.
You build following random variables, based ...
1
vote
5answers
131 views
Probability distribution functions for the perimeter and space of triangle with fixed radius
Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
1
vote
6answers
380 views
Why is the number of possible subsequences $2^n$?
If anyone here is familiar with the Lowest Common Subsequence problem, they probably know that the number of posibble subsequences in a sequence is $2^n$; $n$ being the length of the sequence.
...
0
votes
2answers
98 views
Variance of a function of a normal random variable
I want to define a new random variable $f$ as a function of a normal random variable $v$:
$$f(v)=\begin{cases}C&\text{if } v\ge C\\ \gamma v &\text{otherwise}\end{cases}$$
where $v\sim ...
0
votes
0answers
53 views
How to model mutual independence in Bayesian Networks?
It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations.
Can mutual ...
0
votes
1answer
75 views
geometric distribution throwing a die
The problem says as follows: We throw a die repeatedly. $X$ and $Y$ denote, respectively, the number of rolls until we reach a $5$ and $6$.
Then the question is to compute $E[X\mid Y=1]$ and $E[X\mid ...
-1
votes
6answers
73 views
Density function
Let $f(x) = e^{{-x -e}^{-x}} $ .
How can I check that f is a density function?
I know that it has to be valid that $ \int_{-\infty}^{\infty}{f(x)} = 1 $ , but how to check this?
35
votes
8answers
2k views
What do $\pi$ and $e$ stand for in the normal distribution formula?
I'm a very beginner in mathematics and there is one thing I've been wondering recently. The formula for the normal distribution is:
...
9
votes
3answers
220 views
Are polynomials dense in Gaussian Sobolev space?
Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of ...
7
votes
1answer
1k views
Minimum variance unbiased estimator for scale parameter of a certain gamma distribution
Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f.,
$$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of ...
5
votes
1answer
217 views
For symmetric stable distributions, why is $\alpha \le 2$?
I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact.
Suppose we are trying to come up with stable distributions. From the definition, ...
17
votes
2answers
565 views
Is there a uniform distribution over the real line?
For every interval $[a,b]$, there exists a uniform probability density over this interval, which is the constant function $f(x)=\frac{1}{|a-b|}$ for $a < x < b$, and $f(x)=0$ for all other $x$.
...
6
votes
3answers
235 views
Probability distribution and their related distributions
I am taking a probability course right now and have encountered a lot of interesting distributions and their related distributions.
For example, if we have two independent Poisson distribution ...
5
votes
1answer
177 views
Uniform distributions on the space of rotations in 3D
I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions.
The Haar measure on $SO(3)$.
The uniform ...
3
votes
1answer
1k views
coin tossed until two consecutive heads or tails appear
A fair coin is tossed repeatedly and independently until two consecutive heads
or two consecutive tails appear. What is the PMF of the number of tosses?
edit: in italic
1
vote
2answers
1k views
First exit time for Brownian motion without drift
I am dealing with the simulation of particles exhibiting Brownian motion without drift, currently by updating the position in given time steps $\Delta t$ by random displacement in each direction drawn ...
10
votes
2answers
321 views
very elementary proof of Maxwell's theorem
Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
