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

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0
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0answers
16 views

Maximum-Likelihood estimator [closed]

Given are $N$ Elements, labeled $1, ..., N$. The Elements are randomly shuffeled and then $n$ ($n < N$) Elements are choosen ($x_1, ..., x_n$). How do I found the Maximum-Likelihood estimator ...
0
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1answer
45 views

Geometric Distribution - How to show that a certain event is unplausible?

We have given a geometric distribution with parameter $p$ as well as some result $r$, which we doubt is an outcome of the given distribution. What is the best way to show that $r$ is indeed not a ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
1
vote
1answer
56 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
4
votes
1answer
59 views

Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in ...
2
votes
1answer
14 views

Confusion about non-negative mutual information

The formula I was given for calculating information for a specific stimulus $s_x$ is: $$I(R,s_x) = \sum_i p(r_i|s_x) \log_2{p(r_i|s_x)\over p(r_i)} $$ It was also said that information is always ...
0
votes
1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
1
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2answers
29 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
votes
1answer
30 views

Passing thresholds with uniform random variables

I have encountered a challenging task: I have a bunch of uniform random variables "trying" to pass a certain threshold, and another bunch trying to pass a different threshold, and I need to estimate ...
0
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0answers
42 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
1
vote
0answers
23 views

How to find the conditional probability [closed]

I need to calculate the joint probability $p\left(x,y,z,\dot{x},\dot{y},\dot{z}\right)$ using the following relation: \begin{equation} p\left(x,y,z,\dot{x},\dot{y},\dot{z}\right) = ...
3
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2answers
62 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
0
votes
0answers
21 views

Frequency distribution, notation problem with the classes

I would like to create a frequency distribution. The range of my data is the interval $[a,b]$. I divide this interval into $n$ equal length part intervals (or classes). So each part interval has the ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
31 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
1
vote
1answer
39 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
0
votes
1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
-1
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2answers
29 views

We are making a Bernoulli experiment…

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
1answer
26 views

A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
0
votes
1answer
32 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
3
votes
1answer
50 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
votes
1answer
41 views

What are random variables and its connection with functionals?

Here is an image of the conversation which I had with my Prof. (He's the one in violet and myself in orange) The topic was random variables and other probability related definitions. I tried to ...
0
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1answer
21 views

Multivariate normal conditioned on sum of squares

Suppose that $X_i$ are i.i.d. N(0,1) random variables, and set $S = \sum_{i=1}^n X_i^2$. Then $S \sim \chi^2_{(n)}$, the $\chi^2$ distribution with $n$ degrees of freedom. Compute the induced ...
0
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1answer
11 views

Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
1
vote
0answers
31 views

Poisson distribution given Gamma Distribution

I'm struggling with this one: If $\theta $ is a Gamma$(p,\lambda)$ random variable with $p>1$ and $\lambda>0$. We give the density of the gamma distribution: $ f(x) = \frac { { \lambda }^{ p ...
0
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0answers
17 views

Dirichlet multinominal Likelihood Derivation

In http://www.gatsby.ucl.ac.uk/~edward/pub/inf.mix.nips.99.pdf, equation 10 to 15 Given $$ p(\pi_1, \pi_2,..., \pi_k) \sim Dirichlet(\alpha/k,\alpha/k,...,\alpha/k) = ...
0
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0answers
12 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
0
votes
1answer
17 views

What is the probability density function and cumulative distribution function of x

What is the probaiblity density function and cumulative distribution function of $x$ (where $x\in [\dfrac{-\pi}{2},\dfrac{\pi}{2}]$) such that both $y_1=\sin x$ and $y_2=\cos x$ are uniformly ...
-1
votes
1answer
23 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
0
votes
1answer
24 views

Distribution of the sum of two independent normal variables [duplicate]

Given the two variables $A\sim \text{N}(\mu, \phi^2)$ and $B\sim \text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B\sim \text{N}(\mu + \xi, ...
0
votes
0answers
15 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
0
votes
0answers
56 views

Exponential integral with $x^2$ and $\cos x$

The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips ...
3
votes
3answers
138 views

What's the probability a random number is at least twice as big as another?

Two numbers $m,n$ are chosen from a normal distribution, i.e. the chance that either number lies between $a$ and $b$ is $$\frac{1}{\sqrt{2\pi}}\int_a^be^{-x^2}dx$$ Edit: you could also say ...
0
votes
0answers
28 views

What is expected value of multiplication of two numbers iteratively?

Given some numbers [1..n]. given some intervals [Li..Ri]. and given value space of random variable. Every-time we can choose some value from value-space, and multiply with numbers in interval. How ...
0
votes
1answer
44 views

Orthogonal transformation of multivariate normal

Let $X \sim N_n(\boldsymbol{\mu}, I )$. Let $O$ be an orthogonal matrix, with the first line $\frac{\boldsymbol{\mu}^T}{\|\boldsymbol{\mu}\|}$, and $Y=OX$. It can be proved that ...
0
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0answers
12 views

Non centered Chi-Squared distribution

Let $X \sim N_n(\boldsymbol{\mu}, I )$. Let $O$ be an orthogonal matrix, with the first line $\frac{\boldsymbol{\mu}^T}{\|\boldsymbol{\mu}\|}$, and $Y=OX$. It can be proved that ...
0
votes
0answers
24 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
0
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0answers
12 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
4
votes
2answers
86 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
0
votes
0answers
16 views
1
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1answer
28 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
2
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0answers
22 views

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian. Without loss of generality we may assume that $X$ and $Y$ are ...
0
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0answers
16 views

Does it represent a joint density function

Let equation of a plane be $ax+by+cz = 0$, where $a,b,c,x,y,z$ are random variables. I know the distribution of $x,y,z$, and I need to find the joint distribution of the coefficients $a,b,c$. Let the ...
1
vote
3answers
39 views

Covariance of Binomials

I'm doing some basic error, but I just can't see where... Let $X_i\sim \mathrm{Bin}(\theta_i,n)$, and $X_j\sim \mathrm{Bin}(\theta_j,n)$ I want to find $\mathrm{Cov}(X_i,X_j)$. So, ...
3
votes
1answer
70 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
1
vote
1answer
25 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
0
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0answers
15 views

Urn model over continuous variables

I have a urn containing $p$ liters of an unmixable fluids mixture: $p_i$ liters of fluid A and with $p-p_i$ liters of fluid B. What is the probability that a spill of $m$ liters of liquid, at least ...
1
vote
1answer
54 views

order statistics

Suppose $X_1,..., X_n$ are i.i.d. continuous r.v. with distribution function $F(x)$, and density function $f(x)$. $X_{(1)}<\cdots<X_{(n)}$ are the order statistics. I've already showed that ...
0
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0answers
18 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
0
votes
0answers
98 views

What is expected color of an object?

There is a set of n objects $\{a_1,a_2,\ldots,a_n\}$ each having color $c_1,c_2,\ldots,c_n\}$ respectively. Now a random subset of this objects is selected and painted with some random color from set ...