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

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2
votes
1answer
52 views

What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?

I'm trying to understand the origin of a certain formula used in the solution to the following question: This question relates to the position probability density for a classical particle ...
2
votes
2answers
459 views

Compute variance of logistic distribution

Consider a random variable $X$ with normalized logistic distribution( so that its pdf is $\frac{e^{-x}}{(1+e^{-x})^2}$). It is well known that its variance $V$ equals $\frac{\pi^2}{3}$ but I couldn't ...
2
votes
2answers
132 views

Expected value as integral of survival function

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
2
votes
0answers
95 views

Expectation of Minimum of Multinomially Distributed Random Variables

Assume $I=(I_1,I_2,I_3)$ is a multinomially distributed random variable, say $\mathrm{Mult}(n;p_1,p_2,p_3)$. What is the expectation of the minimum of the first two components $\mathbb ...
2
votes
2answers
6k views

Probability of sampling with and without replacement

In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding ...
2
votes
1answer
249 views

Do Convergence in Distribution and Convergence of the Variances determine the Variance of the Limit?

Suppose we have a sequence $(X_n)_{n\in\mathbb{N}}$ that satisfies: $X_n \rightarrow_d X$, for $n\rightarrow \infty$, where $\rightarrow_d$ denotes convergence in distribution; $\mathrm{Var}(X_n)$ ...
2
votes
1answer
1k views

Distribution of Brownian motion

How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$? I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also ...
2
votes
3answers
667 views

Improper Random Variables

What is an Improper Random Variable? I know the definition in terms of the CDF like, F(∞) - F(-∞) <1. Could any one explain it more clearly, specifically I am looking for an example of an improper ...
1
vote
1answer
29 views

Limit of Multivariate Probability Density Function as one or more or all variables approach positive or negative infinity

Could someone point out a good source or provide justification as to whether the limit of a multivariate probability density function, as one or more or all variables tends to positive or negative ...
1
vote
1answer
65 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
1
vote
1answer
79 views

Expected Value of Maximum of Two Lognormal Random Variables with One Source of Randomness

We have two random variables $X$ and $Y$ which are log normally distributed, with suitable parameters, what is the expected value for $\max(X,Y)$? Given, $$ X=e^{\mu+\sigma Z};\quad Y=e^{\nu+\tau ...
1
vote
0answers
83 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
1
vote
1answer
187 views

Probability of getting k different coupons in a set of S coupons, from an infinite set of N different types of coupons.

This is my slight modification to a question in the book "Introduction to Probability and Statistics for Engineers and Scientists" by Sheldon M Ross. ...
1
vote
1answer
126 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
0
votes
1answer
61 views

Distribution of Square of Rician Random Variable?

We know that the square of a Rayleigh random variable has exponential distribution, i.e., Let the random variable $X$ have Rayleigh distribution with PDF ...
0
votes
1answer
80 views

Probability and Statistics, Penicillin is grown in a broth whose sugar content must be carefully controlled.

Penicillin is grown in a broth whose sugar content must be carefully controlled. The optimum sugar concentration is 4.9 mg/mL. If the concentration exceeds 6.0 mg/mL, the fungus dies and the process ...
0
votes
1answer
248 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} ...
-1
votes
1answer
636 views

Getting $P(X>0,Y>0)$ for a bivariate distribution

$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables. I had to show that $X$ and $Z= {\frac{Y-\rho X}{\sqrt{(1-\rho^2)}}}$ are ...
15
votes
3answers
883 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 ...
9
votes
1answer
220 views

$P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! }$ — What is the name of this probability distribution?

$$ P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! } $$ I'm having a really tough time describing what this distribution does, but it's simple in code. So if you know code, then read on: ...
8
votes
3answers
259 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
7
votes
1answer
289 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
6
votes
1answer
111 views

How can a $\sigma$-algebra be “treated” or computed? Example

My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$. But, imagine now that ...
6
votes
4answers
18k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
5
votes
0answers
3k views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
5
votes
1answer
798 views

Multivariate Hypergeometric Distribution/Urn Problem

I am having a difficulty with the following multivariate hypergeometric distribution problem. The setting is as usual, an urn contains a total of $M$ balls of $K$ unique colors, with $N_1$ balls of ...
5
votes
1answer
5k views

X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$

X, Y are independent standard normal random variables, what is the distribution of $$ \frac{X}{X+Y} $$ Could anyone help me with this? Thanks. I have worked the problem by multivariable ...
5
votes
4answers
1k views

Can you demystify the Power Law?

How would you describe the Power Law in simple words? The Wikipedia entry is too long and verbose. I would like to understand the concept of the power law and how and why it shows up everywhere. For ...
4
votes
1answer
216 views

Random variable exponentially distributed?

I just want to be sure about this: If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ ...
4
votes
2answers
397 views

Characteristic Function and Random Variable Transformation

Let $X$ be a random variable, and let $\phi_X(t)$ be its characteristic function. Let $Y = f(X)$ be a transformation of the random variable $X$ where $f$ is increasing and one-to-one. Is there a ...
4
votes
1answer
435 views

Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?

I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral $$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x ...
4
votes
1answer
7k views

Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that ...
4
votes
2answers
2k views

Is the family of exponential distributions closed under scaling?

While reading wikipedia article on Exponential distribution, I found the statement on scaling the random variable. Let $Exp(\lambda)$ be the distribution of the exponential random variable with ...
4
votes
2answers
1k views

Absolute continuity of a distribution function

This appeared on an exam I took. $Z \sim \text{Uniform}[0, 2\pi]$, and $X = \cos Z$ and $Y = \sin Z$. Let $F_{XY}$ denote the joint distribution function of $X$ and $Y$. Calculate ...
4
votes
3answers
2k views

Integral with Normal Distributions

I know that the following equality is true for any $a$ and $\sigma$ (I have solved it numerically): $$\int\nolimits_{-\infty}^{+\infty}\Phi\left(\frac{a-x}{\sigma}\right)\frac1{\sigma} ...
4
votes
1answer
822 views

Questions about geometric distribution

I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd ...
3
votes
4answers
449 views

Probability for “drawing balls from urn”

I'm afraid I need a little help with the following: In an urn there are $N$ balls, of which $N-2$ are red and the remaining are blue. Person $A$ draws $k$ balls, so that the first $k-1$ are red ...
3
votes
1answer
186 views

Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the ...
3
votes
2answers
79 views

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
3
votes
1answer
155 views

Random sample from discrete distribution. Find an unbiased estimator.

$X$ is a discrete random variable with parameter $a > 0$ whose pmf is defined as: $$ f_X(x) = \begin{cases}0.2, &x = a\\0.3, &x = 6a\\0.5, &x = 10a\end{cases} $$ Say we have a random ...
3
votes
1answer
945 views

Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algebra. Given that $X$ and $Y$ are independent, and that $X$ is independent of $\mathcal{F}$, can I ...
3
votes
3answers
1k views

prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$

How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$. By memoryless property, $$P(X=i+s | X>i)=P(X=s)$$ How to get ...
3
votes
2answers
127 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
3
votes
2answers
757 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
3
votes
1answer
203 views

How to prove these two random variables are independent?

If $X$ and $Y$ are independent Gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$ respectively, how to show that $U=X+Y$ and $V=X/(X+Y)$ are independent?
3
votes
2answers
2k views

Projection of a 3D spherical distribution function in to a 2D cartesian plane

Consider a 3D spherical Gaussian distribution function that depends on radius only, $$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$ where $R_\mu$ is the radial offset of the distribution and ...
3
votes
2answers
4k views

PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
3
votes
2answers
191 views

A variation on the $F$-distribution

If I have $\frac{X/n_1}{Y/n_2}$ where $X$ and $Y$ are independent chi-squared random variables, with degrees of freedom $n_1$ and $n_2$, respectively, then the distribution of this ratio is given by ...
3
votes
2answers
1k views

Find the expectation

A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ? The answer is $\large \frac{ca}{a+b}$. How to approach this one?
2
votes
3answers
59 views

How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf?

I have some problems trying to prove the following problem: A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf ...