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

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416 views

Determining Type of Probability Distribution

Suppose that pollen spores are randomly scattered in a home, at a density of $8$ sports per cubic cm. What is the probability of finding at least two spores in a space of $0.2$ cubic cm? Here is ...
4
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2answers
159 views

Proving that the expectation is always negative

I am interested in if the expected value $$E_{X_3}\left[\log\frac{f_1(y)}{f_2(y)}\right]<0$$ is always be negative or not. Here, I have $3$ random variables $X_1,X_2,X_3$, corresponding to the ...
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2answers
201 views

Number of draws required for ensuring 90% of different colors in the urn with large populations [duplicate]

My problem is: An urn contains $10000000$ ($10^7$) different colored balls, namely $K_1, K_2,\dots,K_n (n=10^7)$ with $K_1=1000, K_2=1000,\dots,K_n=1000$. My question is: How many balls do I need ...
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1answer
114 views

drawing at least 90% of colors from urn with large populations

My problem is: suppose I have an urn containing balls of $n = 10000000$ (i.e., $10^7$) different colors, with $1000$ balls of each color (so the total number of balls is $1000n = 10^{10}$). Suppose I ...
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1answer
206 views

drawing at least one colored ball of each from urn in a case of large populations

My problem is: If an urn contains balls of $10^7$ different colors, namely $K_1, K_2, \ldots K_{10^7}$, and there are 1000 balls of each color, so that the total number of balls in the urn is ...
2
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1answer
705 views

drawing different colored balls from one urn without replacement and at least

I have this problem (numbers in the example are much smaller than reality, so it would help to get a general equation): One urn contains $10$ red, $10$ yellow, $10$ black, $10$ green, and $10$ orange ...
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1answer
105 views

Poisson Process with a Random Variable

I really couldn't wrap my head around this basic concept so I'm looking help for some basic calculations to solidfy my understanding: Suppose we have $T\in (0,\infty)$. Say we have $E(T)=\mu$, ...
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1answer
61 views

Matching pdf with the Inverted Gamma Distribution

So the Inverted Gamma probability density function is: $\displaystyle{f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right)}$ The equation I'm ...
2
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1answer
216 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
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1answer
4k views

Continuous uniform distribution over a circle with radius R

I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x ...
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1answer
372 views

Atomless probability measure

Assume that $m$ is an atomless probability measure on $\mathbb{R}^{d}$. Let $\left( X_{1},\ldots ,X_{d}\right) $ be a random vector with law $m$. Are the marginal cumulative distribution functions of ...
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2answers
67 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
1
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1answer
391 views

Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the ...
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0answers
161 views

Reciprocal Shifted Log-Normal Distribution

Let $X$ be a log-normal distribution, let $k\geq0$ be a real value and let $Y=\frac{1}{X+k}$. What is the name of the $Y$ distribution other than 'reciprocal shifted log-normal'? What is the mean of ...
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1answer
1k views

Exponential Random Variables

QUESTION: Let $X$ and $Y$ be exponentially distributed random variables with parameters $a$ and $b$ respectively. Calculate the following probabilities: (a) $P(X>a)$ (b) $P(X>Y)$ (c) ...
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0answers
2k 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 ...
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3answers
24 views

Distribution of an additional random variable

I need a little help for the following exercise: Let $X_1, X_2,\ldots$ be sequence of iid rv with values in $\{1,2,3 \}$ and $p(i):=P(X=i) \gt 0$ for $i \in \{1,2,3\}$. Define an another rv ...
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0answers
162 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
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0answers
36 views

Convolution of independent distributions

Let F be a distribution on $R$ and X be a randomvariable with distribution F. If $x\geq0,$ $\overline G(x)=P(X>x|X\geq0)$ (i.e. conditional distribution), then ...
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0answers
63 views

Best Distribution to Approximate this Histogram

Given the histogram below, what common distribution would be well fit to this data i.e. Beta, skewed normal. I want to use the data as a prior in Bayesian analysis so want to approximate it by a ...
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1answer
1k views

If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.

Knowing that if you have two independent $X$ and $Y$, and $ f $ and $ g $ measurable functions, how to show that then $ U = f (X) $ and $ V = g (Y) $ are still independent.
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3answers
67 views

Basic statistics - Calculate distribution of winning

I have a 100 sided fair dice with each side labelled 1 thru 100. I win if the number rolled is 49 or higher (1% advantage). 1. What is the probability of me winning exactly 500 rolls if the dice is ...
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0answers
36 views

Holder norm of Wiener process

Show that if $W(t),\ 0 \le t \le 1$\ is a standard Wiener process, then its Holder norm $\sup\limits_{0 \le s,\ t \le 1}\frac{|W(t) - W(s)|}{|t - s|^{\alpha}},\ 0 < \alpha < \frac{1}{2}$\ has ...
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0answers
386 views

Variance of minimum of N random variables

Let $X_1,X_2,\dots,X_N$ be i.i.d. random variables with support $[0,M]$, and with density and distribution functions $f_X(x)$ and $F_X(x)$ respectively. Given $Y= \min_{i\in \{1,\dots,N\}} X_i$ , ...
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1answer
66 views

Need an easy CDF for Inverse transform sampling

I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I ...
0
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1answer
3k views

Expected Values and CDF

I have a piecewise function and I have to find the expected value of $x$ and the cdf. If I have three different pieces for the function, how do I find the expected value? Do I integrate each piece ...
3
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1answer
89 views

Is it possible to sample the Dirac delta function?

The Dirac delta function can be a probability measure with the unit/Heaviside step function as its cumulative distribution function. Is it possible to sample such a distribution? If a random variable ...
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1answer
199 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
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2answers
71 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
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3answers
484 views

How to prove uniform distribution of $m\oplus k$ if $k$ is uniformly distributed?

All values $m, k, c$ are $n$-bit strings. $\oplus$ stands for the bitwise modulo-2 addition. How to prove uniform distribution of $c=m\oplus k$ if $k$ is uniformly distributed? $m$ may be of any ...
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1answer
67 views

Convergence in distribution and normality of the limit

Let $Z=(Z_1,Z_2)$ be a bivariate standard normal vector and $Y_{1,n},Y_{2,n}$ two sequences of real valued random variables with finite variance such that $Y_{1,n}\xrightarrow{d}Z_1$ and ...
0
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2answers
161 views

Probability distribution of the expectation value of a poisson variable given an observed value.

what is probability density $P(\lambda|Poisson(\lambda) = N)$? In other words, if I have a poisson variable $X(\lambda)$, where $\lambda$ is unknown, and I observe $X=N$, what is the probability ...
2
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1answer
163 views

Conditional Mathematical Expectation (problem)

Given that $X \sim CUD(0,1)$ and $Y \sim CUD(0,X)$, where CUD means continous uniform distribution, what is $E(X|Y)$ ? I can't find density function $f(x,y)$. Am I missing something obvious?
2
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1answer
85 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
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1answer
53 views

Does the area under the curve remain the same in this variable transformation?

$X$ is a continuous random variable with probability density function $f(X)$. Let $Y = f(X)$. Let $g(.)$ be the pdf of $Y$. Intuitively I think below relation holds true, how to prove it does or ...
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0answers
144 views

Bayesian updating of multivariate normal?

Let $\bf x$ be an unobserved realization of $\tilde{\bf x}\sim\mathcal{N}(\pmb\mu,\pmb\Sigma)$, where $\pmb\mu\equiv\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}$ and ...
2
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1answer
257 views

Can someone explain the intuition behind this moment generating function identity?

If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= ...
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1answer
399 views

Random vector with uniform distribution.

Let $(X,Y)$ be a random vector with uniform distribution at $0 \leq x \leq 1$, $x \leq y \leq x+h$ with $0<h<1$. Find $E(X)$ and $E(XY)$. What i did: (1) Find densities: $f_X(x) = \left\{ ...
2
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1answer
54 views

Find the probability distribution.

A game is to choose a random real number $x$ between 0 and 10. The earnings are given by $|5-X|$ being X the number chosen. (a) - Find the earning distribution and (b) - If you play twice with $X_1$ ...
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1answer
131 views

How to vary lambda in exponentially distributed numbers

I am implementing an exponentially distributed random number generator (RNG) based on George Marsaglia's Ziggurat algorithm. I previously used the algorithm to create a normally distributed RNG. By ...
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2answers
115 views

Convergence of random variables defined by the normal distribution.

I'm trying to prove this: Given $\{\mu_n\}$ and $\{\sigma_n\}$ sequences of real numbers such that $\mu_n \rightarrow \mu$ and $\sigma_n \rightarrow \sigma$, if $X_n \sim N(\mu_n, \sigma_n^2)$ and $X ...
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1answer
74 views

Bernoulli process failures rate

I have seen an unproved claim, which states that given an infinite Bernoulli process with probability $p$ of success, for every $c<p$, the probability that at any given time the success rate is ...
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1answer
193 views

What is the maximum entropy distribution of points on a sphere that has a fixed non-zero average cosine of the polar angle?

Suppose we have a unit vector in 3D space whose orientation has some unknown distribution $p(\theta,\phi)$. All we know about this distribution is the average value of $cos(\theta)$: ...
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1answer
1k views

Going from the Poisson distribution to the Gaussian.

In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value ...
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1answer
79 views

Joint Distribution applied to quadratic equations.

If i pick random number $b$ and $c$ from $[0,1]$ and then define $p(x) = x^2+bx+c$, what is the probability that p has two real roots?. I've been thinking that it would be enough to know $P\{4c < ...
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2answers
79 views

Probability distribution of the sum of $N$ values from a set of values numbered $1$ to $K$

I have been trying to figure out how to determine the probability distribution function for the sum of $N$ values taken from a set of $K$ consecutive values (valued $1$ to $K$). For example, if I ...
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0answers
87 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
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1answer
2k views

Probability distribution of defective parts

Suppose there are 1 million parts which have 1% defective parts i.e 1 million parts have 10000 defective parts. Now suppose we are taking different sample sizes from 1 million like 10%, 30%, 50%, 70%, ...
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1answer
51 views

Trouble simplifying the pdf of minimum exponentially distributed r.v.

Given the following: $ X_i \sim EXP(1, \eta) $ Asked: show that $Q=X_{1;n}-\eta$ is a pivotal quantity. My approach: $\ \ \ f(x)=e^{-(x-\eta)} \Rightarrow F(x)=\int_{-\infty}^x ...
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1answer
93 views

Gebelein's Inequality and convergence of distribution

We know that for a bivariate standard normal vector $Z=(Z_1,Z_2)$ it holds that \begin{align*} \operatorname{Cov}(1\{Z_1\leq u),1\{Z_2\leq u))\leq \operatorname{Cov}(Z_1,Z_2). \end{align*} This ...