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

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22 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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1answer
26 views

Independent variable vs. Uncorrelated variable confusion. How do I interpret this?

I'm reading Time Series Analysis and Forecasting by Example by Søren Bisgaard and Murat Kulahci and I'm having trouble conceptualizing a particular passage and it's bugging me enough that I can't move ...
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2answers
23 views

Probability and series. (I just need help with the arguments provided )

I have this problem: Suppose that random variable X have possible values $1,2,\ldots,$ and $P(X=j)=\dfrac{1}{2^j}$. Calculate P(X is even). I did the following; I noted that $P(X=2k)=\dfrac{1}{4^k}$ ...
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1answer
32 views

From a stack that contains 25 articles, 5 of them are defective, then we choose 4 at random.

I have the following problem; From a stack that contains 25 articles, 5 of them are defective (20 are not), then we choose 4 at random. Let X be the number of defective articles founded. Get the ...
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2answers
61 views

Difference between two real roots with uniformly distributed coefficents

I have a question that first I need to know what is happening, but then I also need to code it in a program called APPL, which is an extension from Maple18 that I really have never used, yet I have ...
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1answer
19 views

If $x$ is a $\chi^2_{N-n}$ RV. what is $x/N$ as N goes to infinity

We know that if we have $N-n$ gaussian iid RVs $\{e_i\}$ with mean $0$ and variance $1$, the RV $x = \sum e_i^2$ is $\chi^2$ distributed with $N-n$ degrees of freedom. We have $N$ larger than $n$. I ...
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1answer
22 views

distance function between histograms accounting for bucket distance

I'm trying to find a good distance measure for histograms that has the following properties: if histogram A and B have high values in buckets further apart, they're more different than if they had ...
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1answer
35 views

Finding density function, plus showing $X \sim F$ where F is cdf of X, $X = F^{-1}(U)$, $U\sim unif(0,1)$ [duplicate]

Suppose $X$ has a continuous, strictly increasing cdf $F$. Let $Y = F(X)$. What is the density of $Y$? Then let $U \sim unif(0,1)$ and let $X = F^{-1}(U)$. Show that $X\sim F$. The first part seems ...
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1answer
34 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
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3answers
48 views

Combinatorial Distribution with random sample I believe

I have no idea what kind of distribution this is and that is what I would like. Balls are numbered 1 to $N$. We select a sample of $n$ at random. Let $Y$ be the largest number in the sample. Find ...
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1answer
29 views

Poisson and Cumulative Distribution Double Check

this is simply me double checking my answer. Let Y be number of fish caught on a trip with Poisson distribution and $\lambda=4$. What is prob of catching 3 or fewer trout on trip? I said this was ...
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1answer
24 views

Difference of Exponential Random Variables / Linear Transformations of RVs

Suppose X and Y are both distributed exponentially with parameter $\lambda$ and $\mu$ respectively. I am trying to find the distribution of X - Y via this method and it does not seem to be working, ...
2
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1answer
43 views

Negative Binomial

Let $X$ be a negative binomial with parameters $r$ and $p$. So $X$ is the number of trials $k$ till the $r^{th}$ success. My first question is determine which values of $k$ the ratio ...
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1answer
40 views

Distribution of sum of $m$ independent random variables

Let $A_m$ be the sum of $m$ identically distributed random variables that are independent and that have an exponential distribution with parameter $\mu$. How do I prove that $A_m$ has a gamma ...
2
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1answer
21 views

Conditional Probability with Ordered Stats

I am very close to solving this, but this last part is killing me on how to solve it. I dont really know where to begin. Scores run from 0 to 5 with density $f(x)=c(x^2-6x+10)$. An intermediate score ...
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1answer
31 views

Two questions about Almost sure convergence and Uniform integrability

Let $X_n$ and $Y_n$ be two sequences of random variables such that $X_n\stackrel{n}{\rightarrow}C$ almost sure and $Y_n\stackrel{n}{\rightarrow}C$ almost sure, $C\in \mathbb{R}$. Suppose that $X_n$ ...
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1answer
16 views

Use the tail bound to estimate the probability

The heights of trees in a particular forest follow a normal distribution with mean 60 feet and standard deviation 10 feet. The tail bound for the standard normal distribution (i.e. X ∼ N(0, 1)) is: ...
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1answer
88 views

What can we say about $1-F(x) = x$?

$F(x)$ is a probability distribution. Is there any useful characterization of the solution to: $1-F(x) = x$? More specifically, can we say anything about the solution in terms of it's relationship ...
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1answer
21 views

multivariate normal moment derivation

I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$ \mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ...
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1answer
106 views

Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
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1answer
32 views

Prove that an absolutely continuous cdf is continuous

Let $F(x_1,\ldots,x_d)$ be an absolutely continuous distribution function. How to prove that $F$ is continuous? Thank you.
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1answer
32 views

Poisson distribution

Suppose that in a population the probability of survival of an individual would survive is 99/100. If 5 people in the population were to buy an insurance, Using Poisson distribution what is the ...
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1answer
29 views

Distribution with density $x^2\operatorname{exp}\{-x^2/2\}$

I came across the probability distribution with density $$ f(x)=\sqrt{\frac{2}{\pi}}\,x^2\,\mathrm{e}^{-\frac{x^2}{2}},\quad x\geqslant 0. $$ Is this distribution known under a certain name? I only ...
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0answers
13 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
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0answers
24 views

Getting stuck in a loop or the probability of hitting all points in a random walk around a circle.

Suppose you are walking around a circular path made up of $n$ tiles. Each tile $i$ is assigned a distinct value $r_i$ by a random variable uniformly distributed on the set of integers $\{1,...,k\}$ ...
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0answers
13 views

If $X_1,X_2$ are iid U($\theta$,$\theta+1$) find $P(X_1+X_2 > C)$

I need to find the probability, $P(X_1+X_2 > C)$ Where $X_1$ and $X_2$ are U($\theta$,$\theta+1$). The solution I have found gives the following, $\int_{1-C}^{1}\int_{C-x_1}^{1} 1 dx_1 dx_2$ ...
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1answer
26 views

Calculate the hazard rate of the Rayleigh distribution

The Rayleigh distribution is a continuous distribution with one parameter, $\sigma^2$. The pdf of the Rayleigh distribution is: f(x)=($\frac{x}{\sigma^2} )$e^$(\frac{-x^2}{2\sigma^2})$, for x$\ge$0 ...
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1answer
28 views

Finding the joint unconditional distribution of $X$ and $Y=N-X$ for $X\sim \mathrm{Bin}(N,p)$ and $N\sim \mathrm{Pois}(\lambda)$.

The question asks to find the unconditional joint distribution of $X$ and $Y=N-X$, given that $N$ has a Poisson distribution with parameter $\lambda$, and $X$ has a conditional distribution ...
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2answers
14 views

If X~beta($\theta$,1) then Z=-ln(x) ~exp(1/$\theta$)

I keep getting this, $P(Z\leq z)=P(-ln(X) \leq z) = P(X \leq e^{-z}) = \int_{0}^{e^{-z}} \theta t^{\theta-1}dt = t^{\theta}|_{0}^{e^{-z}}=e^{-z\theta}$ $\frac{d}{dz}[e^{-z\theta}]=-\theta ...
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0answers
14 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
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1answer
15 views

Finding the c.d.f.

Two equal rods each of length $2a$ are broken into two parts at points whose positions are random. $X$ is the length of the shortest of the four parts thus obtained. Find the probability, $F(x)$, that ...
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0answers
9 views

Distribution of $Mx$ where $M$ is a Haar matrix and $x$ is a unit vector

I have a $n\times N$ random matrix $R$ with entries i.i.d. standard Gaussian. $n<N$. Then the matrix $M=(RR^T)^{-1/2}R$ is Haar distributed, i.e. has orthonormal rows and uniformly random ...
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0answers
21 views

Understanding Cumulative distribution function

Assume the below probability distribution of a dice \begin{array}{c|ccccc} values & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline p(values) & 0.05 & 0.2 & 0.4 & 0.2 & ...
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0answers
20 views

Mean and variance of Gamma distribution

How do I calculate the mean and the variance of a Gamma distribution? I was told to prove the variance was sigma/lambda(^2), I don't know how to find the variance much less the variance.
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2answers
32 views

Gamma Distribution and Probability [duplicate]

The lifetimes of batteries are independent exponential random variables, each having parameter λ. A flashlight needs 2 batteries to work. If one has a flashlight and a stockpile of n batteries, what ...
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12 views

Inverse-Wishart distribution pdf is different if we derive it directly from Wishart distribution?

According to Wikipedia, there is the following relation between the Wishart and the inverse-Wishart distribution: "If ${\mathbf A}\sim \mathcal{W}({\mathbf\Sigma},\nu)$ and ${\mathbf\Sigma}$ is of ...
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0answers
30 views

What is the maximum entropy distribution over all integers (ie. including negative ones) with fixed mean and variance?

I know that the maximum entropy distribution with over the non-negative integers fixed mean is a geometric distributions. However, I cannot find conclusive information about what are the maximum ...
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1answer
23 views

Distribution of the lifetime of a system consisting of two exponentially distributed components, one being backup

I have a system consisting of components $S_1$ and $S_2$ whose lifetimes $T_1$ and $T_2$ follow the exponential distribution with parameter $\lambda$. At time $t=0$ the component $S_1$ is switched on ...
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1answer
34 views

Is it possible for the median to be left to the mean in the Gamma distribution?

Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew but this is not always true. According to wikipedia, there ...
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0answers
27 views

Conditional distribution on arrival time (Poisson process)

Suppose that $\{N_t: t\geq 0\}$ is a Poisson process of rate $\lambda$ and $T_1< T_2< \dotsb\ $ are its arrival times (i.e. $T_i := \min \{t\geq 0 : N_t \geq i\} $). What is the conditional ...
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1answer
17 views

Best possible approximation of P(X,Y)

I want to show that $\min_Q D(P_{Y|X} || Q | P_x) = I(X;Y)$ and I've arrived at a question. Here $D$ is the KL divergence or relative entropy and $I$ is of course the mutual information. ...
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22 views

Dirichlet parameters influence on samples

If I draw samples from a Dirichlet distribution with parameters $[0.1, 0.1, \dotsc, 0.1]$, I get samples like multinomial, but if I set parameters to $[100, 100, \dotsc, 100]$, I'll get uniform ...
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1answer
66 views

Example of Beta distribution [closed]

Could anyone show a simple example of Beta distibution.How does the normalization takes place in beta distribution?I would like an answer that explains the concepts in laymen terms.
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1answer
50 views

Working out percentages for payment table

I'm here because my mathematics skills are lacking somewhat for a problem i'm trying to solve.. I'm trying to work out a formula for prize distribution for the top 3rd of entrants in a competition. ...
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0answers
17 views

How to find the cumulative distribution function of a sum of n continuous random variables?

Consider the sequence $\{t_n\}_{n\ge 0}$ of independent and identically distributed continuous random variables with distribution function $F(x)$, i.e., $P(t_n\le x)$. Define $~~T_n=\begin{cases} ...
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1answer
10 views

Having trouble trying to find a simple probability for a hazard-rate function?

Suppose that the life distribution of an item has hazard rate function $λ(t)=3.9t^{2}, t>0.$ What is the probability that the item doesn't survive to age 1? Would I just integrate the function ...
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1answer
20 views

Probability of an event if items are drawn simultaneously or not

A box contains 10 balls, with each ball labeled from 1 to 5 (there are two balls labeled with a 1, two labeled with a 2, and so on). 3 are drawn without replacement. What is the probability of drawing ...
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1answer
20 views

Finding parameters that set the variance and skew of a distribution.

Aim I aim to run some numerical simulations where a random variable of interest (Let's call it $X$) can take different distribution. Past studies, always assumed that $X$ is normally distributed ...
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18 views

property of distribution function

Let $f$ a continuous map from $\mathbb{R} \rightarrow \mathbb{R}$ and let $L_1, L_2$ 2 probability measures on $\mathbb{R}$. Let $K$ be a closed set in $\mathbb{R}$. In a proof, I want to use the ...
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1answer
53 views

Can someone explain Wishart distribution?

I have to use Wishart distribution to model noise in images. Can someone explain or give intuition behind wishart distribution. Thank you !!!