# Tagged Questions

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

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### 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 ...
2answers
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### Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.$$
3answers
761 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 ...
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 .. ...
2answers
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### 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 ...
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 ...
1answer
503 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). ...
1answer
366 views

### Limit using Poisson distribution

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
2answers
228 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 ...
3answers
463 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 ...
1answer
98 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 = ...
2answers
331 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 ...
1answer
570 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 ...
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 ...
3answers
249 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 $(-1,1)$?
2answers
496 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 ...
1answer
808 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 ...
2answers
481 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 ...
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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 ...
1answer
729 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: ...
1answer
543 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 ...
2answers
52 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 ...
1answer
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### 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? ...
1answer
238 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 ...
2answers
105 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 $0$ ...
2answers
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### 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 ...
4answers
698 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 ...
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 ...
1answer
278 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 ...
1answer
365 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 ...
3answers
780 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 ...
1answer
224 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: ...
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 ...
1answer
325 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 ...
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 ...
6answers
370 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. ...
2answers
96 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 ...
0answers
52 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 ...
1answer
74 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 ...
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?
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: ...
3answers
218 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 ...
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 ...
1answer
215 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, ...
2answers
550 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 $x$. ...
3answers
232 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 ...
1answer
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### 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 ...
1answer
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### 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
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### 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 ...
2answers
318 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 ...
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