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

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1answer
11 views

When randomly distributing n points amongst m people, what are the odds that one certain person will get a certain amount of points?

I'm mostly curious about how to find this in general, but the actual problem is with 20 points and 5 people. I know probability problems are very counterintuitive, and thus I was unsure after ...
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0answers
14 views

Probability, expected frequency and resultant distribution skewed or not?

A population consisting of a certain proportion of defective items has mean $\mu = 2$. If a sample of 4 items is examined and repeated 200 times, obtain a) probability of an item being defective, ...
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0answers
13 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow ...
1
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1answer
206 views

Conditional expectation of the maximum of two independent uniform random variables given one of them

Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$. What is the conditional expectation of $\max\{X_1,X_2\}$ given $X_2$? And the conditional expectation of ...
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1answer
18 views

Notation: Codomain of a probability density function

I need some help with the correct notation for the codomain of a probability density function. Consider the following problem. Let $$ F : V \to (0,1), \, x \mapsto \int\limits_{\inf V}^{x} f(t) \, ...
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0answers
22 views

A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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0answers
21 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
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2answers
42 views

Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
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0answers
19 views

Probability Help with three events [on hold]

When a piece of information (a bit) is transmitted over a communications channel, it may be wrongly communicated. One method of improving reliability is to transmit the same piece of information an ...
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2answers
292 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
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3answers
27 views

Dice roll - Geometric Distribution Question

I am having a hard time understanding the concept of a negative binomial distribution. For example the question: How many times do you expect to roll a six-sided die before landing on the number ...
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2answers
40 views

What is the distribution of a binomial variable where the number of trials is itself random?

We do the following experiment: Select a random element $k$ from $\{1,\dots,n\}$. Toss $k$ fair coins. Define $X$ = the number of heads. What is the distribution of $X$? Given $k$, the variable ...
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0answers
5 views

Fitting power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
0
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0answers
31 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
0
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0answers
11 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
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1answer
71 views

Probabilities of errors in three independent transmissions

i have been working through some old exam papers and have gotten stuck on this last one. can anyone help? When a piece of information (a bit) is transmitted over a communications channel, it may be ...
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2answers
13 views

Can a geometric random variable have a finite sample space? [on hold]

Can it be finite? I think it has to have an infinite sample space (according to my lecture notes)
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1answer
396 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
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2answers
17 views

Probability Distributions (Tree Diagram)

Satish picks a card at random from an ordinary pack. If the card is ace, he stops; if not, he continues to pick cards at random, without replacement, until either an ace is picked, or four cards have ...
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3answers
32 views

Finding the Marginal Distribution of Two Continuous Random Variables

The continuous random variables $X$ and $Y$ have the joint probability density function: $$f(x, y)= \begin{cases} \dfrac{3}{2}y^2, & \text{ where } 0\leq x \leq 2 \text{ and } 0 \leq y ...
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0answers
25 views

Proof that Sum of $n$ Squared Errors ~ Chi Square with $n$ $df$

There is a youtube video dealing with the proof that the sums of the squares of normally distributed $n$ random errors, each one distributed as $\sim \chi^2(1\text{ df})$ follows a chi square ...
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2answers
38 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
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1answer
56 views

Breaking probability theory by having a different number of random variables depending on a conditioning random variable.

I suspect I'm breaking probability theory but I don't know how or why. How does one handle working with conditional probabilities where one can have a different number of random variables depending on ...
1
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1answer
45 views

A coin probability question

Let $p$, $q$ be values in $[0,1]$ and $\alpha \in [0,1]$. Assume $\alpha$ and $q$ known, and that $p$ is unknown parameter we would like to estimate. A coin is tossed n times, resulting in the ...
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1answer
18 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
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0answers
16 views

Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na ...
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0answers
9 views

Estimating Johnson Distribution Parameters by Quantile

In this paper: http://www.researchgate.net/publication/31291960_Quantile_Estimators_of_Johnson_Curve_Parameters the four parameters for the Johnson distribution ...
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2answers
24 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
0
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1answer
29 views

Functions of a random variable

Assume that $Y$ ~ $Exp(Ω)$. Find the cdf and pdf of $Z$ = |$Y$ - $δ$|. In order to solve this question so far, for $Y$, I am thinking about using the pdf equation for the exponential distribution i.e. ...
2
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1answer
28 views

Poisson Approximation of Binomial

I have to prove the Poisson approximation of the Binomial distribution using generating functions and have outlined my proof here. Given, \begin{align} & \lim_{n\to \infty} np_n = \lambda \\ ...
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0answers
47 views

Need simplified formula of probability equation

I have RV $x$ which is function of independent continuous RVs $x_1$ and $x_2$. After some manipulations, I came up with an expression for the outage probability of $x$ as $$P_\text{out}(y)=\Pr(x\leq ...
2
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1answer
30 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
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0answers
34 views

Sufficient condition for convergence in distribution in the plane

I'm trying to show convergence in distribution for a sequence $X_n$ of random variables in the plane. Here's what I know. I have a sort of squeeze theorem for the probability of the r.v.s being in a ...
1
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1answer
19 views

If $x_1, \ldots, x_n$ have probability distribution function $F(x)$, then the maximum has probability distribution function $F(x)^n$

A random sample $x_1,x_2,.....,x_n$ is taken from a population , which has the probability distribution function $F(x)$ and the density function $f(x)$ . The values in the sample are arranged in ...
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2answers
152 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
2
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1answer
378 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
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2answers
39 views

A trivial question about prediction of arrival rate of a Poisson process from sample data

A bus arrives at a bus stop according to a Poisson process. It is given that in the last 100 hours, the bus arrived at the bus stop exactly 200 times. Predict the arrival rate for the bus at the bus ...
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2answers
27 views

How to proof that the median of a lognormal distributions equals $\exp(\mu)$ [on hold]

If $V$ is lognormal distribution, how can you prove that his median equals $\exp(\mu)$? With $\mu$ the mean of the normal distibuted $\ln(V)$
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1answer
12 views

If $E[(X \wedge b) \vee a] = E[(Y \wedge b) \vee a]$ all $a\leq b$ is $X =^d Y$?

If $X,Y \in L^1$ and $E[(X \wedge b) \vee a] = E[(Y \wedge b) \vee a]$ for all $a\leq b$ do we necessarily have $X =^d Y$? Taking $a=0$ the integrands are positive so we may use fubini to find ...
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0answers
16 views

normal distributions

Let $B$ be a random vector with $b_{i} \backsim N(m_i,\sigma_i) $ and $Y$ another random vector with $y_i \backsim N(r_i,\psi_i)$. Let $A$ be a symmetric and non-singular square matrix. What is ...
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0answers
6 views

How to test if an number generated from an exponentially distributed random variables is statistically different from zero at some confidence level?

For instance, let the generated number be 0.3 and the mean of the exponential distribution where this number comes from be 0.03. At a significance level of 0.10 or 0.05, is it 0.3 statistically ...
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0answers
12 views

Bernouilli-distribution, geometric-distribution conditional probability

A homework question I'm stuck with: Let X and Y be two independent discrete random variables. X has the Bernoulli-distribution with parameter $\alpha$ : $$ p_{X}(0)=1-\alpha, p_{X}(1)=\alpha $$ Y has ...
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1answer
26 views

The CDF and PDF of the transformation of a random variable (absolute value) [on hold]

Let X~Exp(λ). Calculate and find the CDF and PDF of Y = |X-μ|. So far my working on paper is here, but I get stuck on how to continue. Any suggestions would be greatly appreciated! ...
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1answer
22 views

Explanation for “jointly pdf is constant but marginal pdf is not”

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$ From short computation, we know: Jointly pdf: $f_{XY}(x,y) = 2$ Marginal pdf of $x$: $f_{X}(x) =\int_0^x ...
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4answers
101 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
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0answers
12 views

Kullback leibler divergence between two language models

My question is associated with comparing two n-gram language model using KLD. Consider a 2 bi-gram language models: $P(S)= \prod_{i-1}^{l} p(w_i|w_{i-1})$ and $Q(S)= \prod_{i-1}^{l} ...
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1answer
1k views

Joint cdf and pdf of the max and min of independent exponential RVs

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
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0answers
38 views

How to find the distribution from a given form of generating function

I have the generating function defined by F(x)= $\sum P(n,s) x^n$ . And the expression for F(x) is given by $F(x)= e^{\frac{a}{b}x} (1-x)^{b/s}$. Then how can i find p(n,s) function..? Can anybody ...
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0answers
15 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
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1answer
555 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...