# Tagged Questions

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### Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
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### Dynamic Biased Randomized Selection Algorithm

I'm looking to implement an algorithm that picks items from a list randomly, albeit in a biased way. Say I have a value priority for each element in that list. I would like an element with higher ...
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### Three person simultaneous random walk

So let's say you have 3 people walking 100m, from one wall to another. Each move each person independently draws 3 integers, each between -10 and 5 with equal probability. You, as the coordinator, ...
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### Random Functions

In computers or to be more specific programming, I can call for a "random" function which generates "random" numbers. How does it do this? Is there a mathematical function that produces eratic and ...
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### Unit random process

My friend asked me to help with the problem on the random processes, but I am stuck as well, because I don't understand the notation $X_t = 𝟙_{[U,1]}(t), t \in [0,1]$ Could anyone explain this one ...
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### zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
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### fidi of Chi-square Random Field

The $\chi^2$ random field $U(t)$ with $n$ degree of freedom (dof) is defined as: \begin{align} U(t) = \sum_{i=1}^n X_i(t)^2, t\in\mathbb{R}^N \end{align} where $X_1(t),...,X_n(t)$ are i.i.d ...
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### What is exactly meaning of The notion of Sample path and Stochastic Process?

I am wondering what Stochastic Process is exactly meaning. Let me talk about what I understood. I will give an example. $\Omega_i$ is noise of my robot's circuit on July the $i$-th day. The ...
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### More concise $f(x) = (\mathop{\mathrm{rand}}(20)-10)\times 10^{\mathop{\mathrm{rand}}(2x)-x}$?

Assume a typical (I think) PRNG $\mathop{\mathrm{rand}}(n) = \omega$ where $\{\; \omega \in \mathbb{R} \mid 0 < \omega \le n \;\}$ for $n > 0$. I want to create a random function $f(x)$, such ...
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### What's the name of this extremely common but extremely pathological continuous function?

Okay, so let's define a random function $F$, such that the value of $F(x)$ is uniformly distributed on $[-1,1]$, and such that for any $x$ and $y$ with $x \ne y$, $F(x)$ and $F(y)$ are independent. ...
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### Invertible pseudo random number generator

I want to create a random sequence using a given seed. However when given a sequence I also want to calculate the seed which produces the sequence. Of course this is not possible using a "true" ...
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### In writing a simulator to simulate an experiment that rolls 2 dice and checks if the sum of the 2 rolls is less than or equal to a given value.

Is it better to use 2 independent random number generators or one array of size 36 that holds the sample space(of all possible sums) and use one random number generator to choose from this arry. ...
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### Mathmatical notation of random function

A function $f$ projecting from $\mathbb N$ to $\mathbb N$ is denoted as $f: \mathbb N \rightarrow \mathbb N$. I is OK to denote the common random() function, i.e., without input parameters, as it is ...
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### PRNG Improvements

Purpose This is the (somewhat) mathematical representation of an algorithm for a pseudo random number generator. It uses mostly linear math and generally is not very complex, but then again - I'm not ...
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### Random number function (counting)

I have task I can't get my head around, even with a suggested answer. You have a function the generates a random integer between $0 - 65535$. Your task is to generate random integers $125-525$ ...
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### Generate random number according to any equation

So I'm after a random number generator where the probabilities of a number occurring in some range is matched to some function. Only really looking at functions with nice integrals (for simplicity ...
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### Uniform random number generation

Given a uniform random number generator that generates integers between any given range, a n-tuple $b$, and an integer $c$, how can one uniformly generate n-tuples ($x$) that meet the following ...
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### How to find the distribution of a function of multiple, not necessarily independent, random variables? [closed]

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
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### The autocorrelation function of i.i.d process

$\{x(n)\}$ is i.i.d; therefore, it is strictly stationary. Can I say the autocorrelation function $\{x(n)\}$ is a delta function, that is $R_X[k] = N_0\delta(k)$? Thanks
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### Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
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### finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
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### Is a function with a random variable continuous?

I often like to fool around on graphing calculators when I am bored. A function that can be very amusing is $f(x) = rand \times \sin x$ Now, on my TI-84 Plus, this looks obviously discontinuous ...
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### The “size” of a continuous uniform selection of points in the unit square

Let $\{X_r\}_{r\in[0,1]}$ be i.i.d. random variables, each distributed uniformly on $[0,1]$. Let $S\subseteq[0,1]^2$ be the random set defined as follows: $$S=\{(r,X_r)\mid r\in[0,1]\}$$ How would ...
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### Finding functions where the increase over a random interval is Poisson distributed

I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function g(t)=f(0,t)...
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### Random processes: Repair time

I have a question that is to do with qeueing theory and repair times: Assume that a small office has 4 printers. Each printer breaks down independently of the other printers and independently of the ...
I have a question if the following relations on conditional probabilities hold for independent random variables? $$P_{X \mid Y, G(Y)}(x_1)=P_{X \mid \{Y\}}(x_2)$$ where $G$ is not necessarily ...
I have a question about conditional distribution. Suppose we have three independent random variables $X_1$, $X_2$, $X_3$. Then we have mapping $Y_1=g(X_1, X_2)$. The mapping is not necessarily an ...