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

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1answer
82 views

How to find the distribution of x-y considering inverse guassian

Let X and Y both be distributed Inverse Guassian which are independent, what is the distribution of Z=X−Y? is there any closed form for distribution of Z?!
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1answer
30 views

How does one go about finding a distribution for this property of the distribution?

I am told I need to find a probability distribution in which this Chebyshev Inequality is fulfilled: $P(|X-\mu|\ge 5\sigma)=.04$. What I tried was just taking a simple distribution where the support ...
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0answers
65 views

What does a partition function tells you?

I have some difficulties to really understand what does a partition function say about your data/observations. For instance, say that we have a price series $P(t)$ on the time interval $[0,T]$ and ...
1
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0answers
137 views

KL divergence of multinomial distribution

Consider $q(x)$ be a Multinomial distribution over $\{1, \ldots, k\}$ with parameters $\{\theta_1,\ldots, \theta_k\}$. And p(x) over $\{1,\ldots, k\}$ with distribution $p(x)=\frac{1}{k}$. Then what ...
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3answers
114 views

What should be the proportions of a three sided coin?

A classical coin has almost no chances of ending its course on the side when tossed. A round pencil with both ends flat has no chance of ending its course on the tip, when tossed. What would be the ...
1
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2answers
63 views

How could I predict the probability of admission/rejection based on prior admissions/rejections and scores?

Assume I have a large number of data points (admitted, score) representing applicants who have either been admitted or rejected and who have some score. Assume we know there is a positive correlation ...
2
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0answers
40 views

How to change variables of a probability distribution when variables have different dimensions

Is this even possible? Say I have a pdf $p(x,y\mid\phi)$, and a function $z = f(x,y)$. Is there a way to derive $p(z\mid\phi)$? The usual change of variables rule $$p(z\mid\phi) = p(x,y\mid\phi) ...
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2answers
393 views

Multivariate Hypergeometric Distribution With Wildcard

In the wikipedia link http://en.wikipedia.org/wiki/Hypergeometric_distribution it is obvious you can do things like "I draw 5 cards from a deck of 50 cards where there are 10 cards that equate to ...
1
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1answer
28 views

Is there a distribution like this?

Is there a distribution like in the picture? It don't need to be the same, but like the idea (postive mean, negative next to mean and zero against $-\infty$ and $\infty$).
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1answer
72 views

What type form of CDF fits this graph?

Hi all, what kind of functional form of CDF do you think closely resembles this shape? Thanks!
1
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0answers
29 views

Product of numbers and gaussian function

Trying to approximate a gaussian function $g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right)}$ with another function I found the product ...
3
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2answers
848 views

Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable? ...
2
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0answers
142 views

Differentiability of a continuous (not absolutely continuous) CDF

Similarly to Cantors Distribution define a Cumulative Distribution Function $F$ on $[0,1]$ as follows: Let $\mu$ be the measure on $\{0,1,2\}$ with $\mu(\{0\})=2/5, \mu(\{1\})=1/5, \mu(\{2\})=2/5$. ...
3
votes
1answer
92 views

The time of waiting at a bank.

There is a bank. The time expressed in minutes that every customer took to finish his job at the bank is given(the numbers are distributed with exponential distribution): $7.4,\ 7.5,\ 8.5,\ 29.2,\ ...
0
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1answer
71 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
2
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3answers
133 views

Bernoulli trials required for k successes

What is the expected value of number of Bernoulli trials required for k successes? Assume probability of success in a single trial = $p$, probability of failure = $q = 1 - p$. I managed to derive the ...
1
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1answer
2k views

Calculating the expected value and the standard deviation of the total profit.

Company sells 2 kind of cars, Lamborghinis and Ferraris, and the sales of the cars are independent. From every sold Lamborghini the company gets 10000 dollar profit and for every Ferrari the company ...
0
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1answer
309 views

Independent and Identically Distributed and the Discrete Uniform Distribution

I am learning about IID. I would like to know from expert if it is accurate to say that the Discrete Uniform Distribution is actually an example of IID random number generator? (if is accurate to say ...
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2answers
77 views

What is the distribution of $c^x$? ($c$ is a constant, $x$ is a random variable)

What is the distribution of $c^x$, where $c$ is a constant and $x$ is a random variable? For example, $x$ follows a Poisson distribution, what is the distribution of $2^x$?
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3answers
689 views

Cumulative distribution function, integration problem

Given the continuous Probability density function $f(x)=\begin{cases} 2x-4, & 2\le x\le3 \\ 0 ,& \text{else}\end{cases}$ Find the cumulative distribution function $F(x)$. The formula is ...
3
votes
2answers
510 views

PDF of summation of independent random variables with different mean and variances

What can we say about the probability distribution function of $n$ independent random variables with different means and variances but the same PDF? For example, lets say $X_1,X_2,\dots,X_n$ are ...
0
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1answer
155 views

A gardener plants three maple trees, four oaks, and five birch trees in a row. …

The question and solution is taken from here. Question: A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally ...
2
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3answers
136 views

expected value of a $e^{-2|x|}$

Let $f(x)=e^{-2|x|}$ Find: $E(X)$ $E(|X|)$ $E(x')$ where $x'$ denotes the largest integer not greater than $x$. I'm stuck on this question and am confused about how to use the modulus sign. I ...
1
vote
1answer
81 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
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0answers
37 views

Is there any relation between cdfs as in set theory?

This might be a silly question, but I could not find any useful information about it. Take two uniform distributions, $F\sim\mathcal{U}(-1,2)$ and $F^{\prime}\sim\mathcal{U}(0,2)$. Is there some ...
0
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1answer
104 views

probability density and distribution

Let $X$ and $Y$ be uniform random variables where $0\leqslant x,y\leqslant 1$. Let $\operatorname{sgn}(x)$ be $1$ when $x>0$, $−1$ when $x<0$ and $0$ when $x=0$. Find the distribution and ...
2
votes
1answer
112 views

Discovering the joint distribution for two dependent random variables?

Suppose I have three continuous random variables $X_1$, $X_2$, and $Y$, where $Y = X_1+X_2$, and $X_1$ and $X_2$ are dependent. If I know the probability distributions separately for $X_1$, $X_2$, ...
0
votes
1answer
36 views

Density derivatives problem

Following problem has been bothering me for a bit. Imagine a n-dimensional vector V and its density defined as $$ f(V): R^n \rightarrow R. $$ I would like to calculate the derivative of the ...
3
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1answer
90 views

Prove expectation inequality

Any ideas on how I could prove the veracity or falseness of the following inequality? Let $X:\Omega \to \mathbb{R}$ a random variable such that the expressions under are well-defined. Then $$E[e^X] ...
0
votes
1answer
254 views

Gibbs sampling to produce posterior pdf

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all $i = 1, 2, ...
0
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1answer
41 views

On uniform number generation with vectors

Let $\vec{a}$ be a random unitary vector. If $\vec{\lambda}$ is a uniformly distributed vector on $\mathbb{S}_2$ (the unitary sphere?), could we say that the result $|\vec{a}.\vec{\lambda}|$ is ...
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2answers
62 views

How to find this limit in statistics?

$$ \lim_{n \to \infty} \left( \frac{n}{2}\cdot\left(y - t + n^{-1}\right)\right) $$ where $y$ is a random variable and $t$ is a real number. Its result seems to be infinity but my book said that it ...
1
vote
1answer
140 views

What are moments and why do we need them?

I am learning probability and have a question about moment generating functions. So far I've learned how to calculate them and how to find the nth moment of a probability distribution. My question is, ...
1
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1answer
99 views

Is the following property for positive random variables fulfilled in general?

Suppose we have a continuous random variable $X$, defined on the interval $[0, \infty)$, which has density $f(x)$ and a finite expectation and variance. I am wondering whether the following is true ...
0
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1answer
70 views

Creating a function from known data and variable relationships

I'm developing a game and I need to create a predictable function while most of the variables are not 100% under my control. I will explain the practical situation: You have two characters, trying to ...
1
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0answers
48 views

Notion of Distribution

I have some difficulties in understanding the notion of distribution. As I understand, distribution, is some source of data, where each element of the data have some probability to occur, for example ...
1
vote
1answer
101 views

Probability using volumes wedge

Suppose that a point $(X, Y, Z)$ is chosen uniformly at random from the wedge $f(x ,y,z)$ belongs to $\mathbb{R}^3: 0 \leq x, y \leq 1, \textrm{and}\, 0 \leq z \leq x$. Compute the probability $((a ...
3
votes
1answer
188 views

Non-convergence of Cauchy Random Variables

Suppose $X_1,X_2,\ldots$ is a sequence of Cauchy random variables with density $$f(x)=\frac{1}{\pi(1+x^2)}, \hspace{3mm}x\in \mathbb{R}$$ and let $S_n=X_1+\ldots+X_n$. It's easy to show that ...
1
vote
1answer
88 views

Issue with a Poisson process and its jump times

Let $(N_t)_{t\geq 0}$ be a Poisson process and $$T_n = \inf\{t\geq 0, \ N_t \geq n\}$$ Now given $t \ge 0$ how to compute $$ \mathbb{E} \left[ \sum_{n=1}^{N_t} X_{T_n}\right] $$ ? where $(X_t)_{t\ge ...
5
votes
1answer
4k views

sum of two independent exponential distribution

let $Y_1\sim \exp(\lambda_1)$ and $Y_2\sim \exp(\lambda_2)$ and $V=Y_1+Y_2$ Show that the pdf of $p_V(x)$ of $V$ has the following form ...
1
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1answer
158 views

Distribution of sum of mixed random variables

Suppose there are $n$ i.i.d mixed random variables $U_1,U_2,\cdots,U_n$. Each has a mass of probability $e^{-\tau}$ at $0$ and a pdf $$f(u)=\left\{\begin{matrix} e^{-u}, & 0<u\leq \tau\\ 0, ...
2
votes
3answers
437 views

Random number generator with discrete probability distribution

Is there a general algorithm for implementing a PRNG with a probability distribution?
0
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1answer
119 views

Analytic solution of the convolution of two discoutinous c.d.f s

I have a c.d.f of variable X with a mass point at the end point, $$F(x) = \begin{cases} 0 & x<a,\\ 1-\frac{m}{x+m-a} & a\le x < r-a,\\ 1 & x\ge r-a. \end{cases} $$ where m>0. Is it ...
3
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0answers
185 views

Calculating expectations in terms of quantile functions?

I have a well behaved random variable, $X$, where I can solve for the quantile in closed form, but in general cannot invert it to get the pdf/cdf. Assume whatever you need on the properties of $X$ ...
1
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1answer
165 views

Convergence to a non-degenerate distribution

If we have iid observations $\mathbf {(X_1,Y_1),(X_2,Y_2),\dots}$ from bivariate distribution $G$ supported on unit disc $\mathbf {[(x,y): 0 \le (x^2,y^2) \le 1]}$. Suppose that distribution has a ...
2
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0answers
273 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
2
votes
1answer
266 views

Is there an analytical solution to the integral of Weibull cdf $1-e^{-(x/a)^b}$

As part of a bigger exercise I need to try and find the integral of $$1-\exp\{-(x/a)^b\} dx.$$ Note: I can't quite get LaTex to format the equation properly but the exponential should be raised to ...
3
votes
1answer
336 views

Distribution of the minimum of two random variables

Here is my problem. Let us consider X, Y and Z to be random variables following the exponential distribution (same mean or not does not matter). I am trying to find the distribution of the ...
0
votes
1answer
147 views

discrete uniform distribution

if discrete random variable X is uniformly distributed over {-7,-5,-3,-1,1,3,5,7},then how to calculate the expectation of X and mod(X) and also expectation of X^2 and mod(X^2).It would be appreciable ...
2
votes
1answer
108 views

Random processes: Repair time

I have a question that is to do with qeueing theory and repair times: Assume that a small office has 4 printers. Each printer breaks down independently of the other printers and independently of the ...