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

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2
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3answers
639 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
58 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
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1answer
77 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
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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
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1answer
158 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
125 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 ...
1
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2answers
1k views

compound of gamma and exponential distribution

What is the distribution of a exponential distribution, whose parameter is drawn form the gamma distribution $ X \sim Gamma(\alpha,\beta)$ $ Y \sim Exp(X)$ how is $Y$ distributed? EDIT Since there ...
1
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1answer
147 views

Help with understanding the $\chi^2$-distribution

I'm studying statistics and there's one part in my book I can't understand. I tried to make as good translation as I can of the problematic part...here goes: Chi squared $\chi^2$ distribution Let ...
1
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1answer
68 views

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive ...
0
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1answer
32 views

An application of the Inverse function theorem

I have recently come across two formula's that I am unfamiliar with and would like to know if they are both aspects of the same thing: ...
0
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1answer
67 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
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1answer
44 views

calculating the probability of k changeovers when flipping a coin

Consider n independent flips of a coin having probability p of landing heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance,if n = 5 and the outcome ...
0
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1answer
211 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*} ...
0
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1answer
70 views

Joint probability generating functions, help please!

With a sequence of $N$ independent Bernoulli trials performed, where $N \in \mathbb{Z}^+$ and the probability of success on any trial is $p$, and $S$ and $F$ being total number of success and fails ...
0
votes
2answers
1k views

Convolution of continuous and discrete distributions

Assume we have two random variables $X$ and $Y$, such that $X \sim P(x)$ and $Y \sim G(y)$. We ask, what is the distribution of $Z = X+Y$. If both of the distributions of $X$ and $Y$ are discrete, ...
10
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2answers
419 views

Does variance do any good to gambling game makers?

People always like to evaluate the variance, but is there any way for variance to be interesting to the gambling game makers? In another word, what is a pratical gambling game that involving some ...
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
252 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
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1answer
279 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
6
votes
1answer
110 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
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1answer
157 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
6
votes
4answers
17k 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
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0answers
184 views

Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
5
votes
1answer
773 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
0answers
51 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
4
votes
3answers
197 views

Intuition behind Variance forumla

Variance is given as: $\operatorname{Var}(X) = \mathbb{E}[(X-\mathbb{E}(X))^2]$. Is there an intuition behind this and can you find this formula starting from the second generating moment ?
4
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1answer
207 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
4answers
876 views

Why does the Cauchy distribution have no mean if it's symmetric around 0?

Something that didn't make intuitive sense to me when learning about the Cauchy distribution was that there was no defined mean for the function, even though the function was clearly centered at zero ...
4
votes
2answers
365 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
425 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
3k views

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed?

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed? Any explanation would be very appreciated.
4
votes
2answers
8k views

Cumulative distribution function of sum of binomial random variables

I was wondering how to get the cumulative distribution function of a sum of two random binomial variables. X + Y, where X has n=15 trials and Y has m=15 trials and probability=0.2 for both ...
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
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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
808 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
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2answers
77 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
142 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
856 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
2answers
203 views

Mean of an increasing function over exponential distribution

I came across the following problem in my research I have two random variables $X, Y$ which are exponentially distributed and $Y$ has a higher mean than $X$. Then I have a function, say $f(z)$, ...
3
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0answers
169 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
3
votes
2answers
718 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
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 ...
2
votes
0answers
50 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
2
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1answer
87 views

Covariance between previous and next occurrence.

Consider a Poisson process where the time between occurrences is random from a distribution with density function $f(t)$. Assume that we are at a random point in time $T$, so likely between two ...