# Questions tagged [random-functions]

This tag is for questions relating to the functions of random variables which is a function from $Ω$ into a suitable space of functions (where $Ω$ is the sample space of a probability space that has been specified). Technically, there is also a measurability condition on this function.

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### Expected value and variance for a homogeneous polynomial of random variables

I want to obtain equations for expected value and variance for a homogeneous polynomial of arbitrary order $n$. For a quadratic form, which is a special case of a homogeneous polynomial, there are ...
0answers
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### Solving an implicit equation with random parameters

I came across the following implicit equation for a real quantity $X$: $$G(X) = X - 1 - \dfrac{a}{n}\sum_{i=1}^n \dfrac{1}{X - b_i} = 0$$ where $a$ is a constant and the $b_i$ ($i = 1,\dots, n$) are ...
0answers
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### Integral analysis of random variable

Need help in solving the $Pr(\frac{xy+ayz}{xz+ayz}<b)$ where x,y,z are exponential random variables with parameter $\alpha$, $\beta$, $\delta$ and a and b are constants. How do I solve this ...
1answer
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### Probability distribution of a function of two random variables

I'm trying to find the probability distribution of a function $q(x, y)=x^2+y^2$, where $x$ and $y$ are normally distributed: $x$~$N[0,\sqrt{V_x}]$ and $y$~$N[0,\sqrt{V_y}]$. I began by finding the ...
1answer
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### Highly Random Function

Call a function $f\colon\mathbb{R}\to\mathbb{R}$ highly random if: Say $T$ is a Turing machine which attempts to compute values of $f$. Given enough values to compute, the cumulative error of $T$'s ...
0answers
47 views

### optima of a random function / stochastic process

Consider the probability space $(\Omega, \mathcal{F},\mathbb{P})$, and let $X:\Omega \mapsto \mathbb{R}^T$, where $T$ is an index set, be a random function. What is the canonical definition of the ...
0answers
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### Is it possible to manipulate a random stream?

I got two random streams of binary data $A$ and $B$. During each step, I can choose to pull a byte from stream $A$ or stream $B$. My goal is to reduce the chances of pulling a pattern* $P$. Is there a ...
0answers
38 views

### How to compute E[$g(X)$] in terms of $X$'s CCDF (complementary cumulative distribution function)?

If we know the PDF $f$ of a random variable $X$ then we can compute an expression like $\mathrm E[g(X)]$ as $$\mathrm E[g(X)] = \int_{\mathrm{Im}(X)} g(x) f(x) \mathrm dx \, .$$ Let $F$ be the CDF of ...
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1answer
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### Is there a simple reduction from permutations of {1, … 2M} to {1, … M}?

Suppose I have a random permutation uniformly chosen from the set of all permutations of $M N$ elements; often in the contexts that I am interested in, this is going to be $2^{m+n}$ elements. And let ...
0answers
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### Continuous variable function that evaluates to random curves

There exists a function of a continuous parameter function that sometimes evaluates to one curve and other to a different one? For instance, is there a function $f(x)$ that say is 0 for $x < 0$ and ...
0answers
8 views

### Bound the error for approximating $g(Z) \approx g(h)$, where $Z \sim N(h, \sigma I)$ and $g$ is high-dimensional

I have multivariate Gaussian RVs that have the following properties: $Z \sim N(h, \sigma_Z I)$, where $h$ is a deterministic vector, $I$ is the identity matrix. $X = g(Z) + N_x$, where $g$ is a ...
1answer
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### Relationship (sum/difference) between elements having normal distribution

I have a set of elements $\{x_1, x_2,...,x_n\}$ having normal distribution. Now, I want to choose $K$ random pairs $(x_i, x_j)$ and compute their difference $\Delta x = |x_i-x_j|$. How can I compute ...
2answers
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### Expected value of a transformation

If $X$ is a continuous random variable with $EX = \mu < \infty$ and $Y = \exp(a|X|)$ for some $a > 0$ Is $EY < \infty$. How might one go about confirming this? Is knowing the distribution ...
2answers
66 views