# Questions tagged [random]

Questions relating to (pseudo)randomness, random oracles, and stochastic processes.

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### Existence of stochastic process with square-integrable realizations

Consider a stochastic process $X = (X(t))_{t\in\mathbb{R}^n}$ with $n\geq 1$ and each $X(t)$ is a real- or complex-valued random variable with \begin{equation*} \mathbb{E} ( X (t)) = 0 \quad \text{and}...
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### Bounding the quadratic form of a random matrix

I encounter a problem of bounding the smallest eigenvalue of the following $p\times p$ matrix: $$A=\sum_{j=1}^{n}Y_jY_j',$$ where $Y_j=n^{-1}\sum_{t=1}^{j}x_t+n^{-1/3}\psi$. Here $\{x_t\}$ is a ...
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### Does a truly random sequence in the range of x..y average to the average of x and y?

I recently started wondering if the average of a truly random sequence between numbers x and y has to be the average of x and y itself. This little JavaScript function seems to prove it (uses ...
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### Probability problem on Unif(0,1) random variables

Given three iid Unif(0,1) random variables a, b, and c, what is the probability that c is the maximum if a+b < 1? The probability distribution of a+b for a+b < 1 is a+b and for 1 < a+b < 2 ...
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### Approximate way to find function of multiple random variables

I want to know if there are approximate solutions to the joint distribution of functions of random variables, where the variables can be only normal or lognormal. The functions would be using only ...
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### Selecting a random number from $(0,1)\cup(1,\infty)$

A random number $a$ is selected from $(0,1) \cup (1,\infty)$. Let $A$ be the event that (a random number from $(0,1) \cup (1,\infty))$ $\in (0,1)$. Let $P(A) = x$ Let $B$ be the event such that (a ...
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### Sampling from Haar measure [duplicate]

Is there an easy algorithm that outputs a random element of $SO(3)$ distributed according to Haar measure on $SO(3)$? Or more generally, replace $SO(3)$ with any compact matrix Lie group. Note: I'm ...
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### Pseudo random ordering of integers

I remember an old retro effect for a screen resolution of $320\times 240$. You would iterate the pixels in a linear fashion so there are $76800$ pixels. You could iterate then one by one starting at ...
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### How is the sample space of a random variable defined?

I was watching this example where the professor said that if we have 2 i.i.d RVs, both of them being binomial random variables, then we can only add them up if their sample spaces are the same. I am ...
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### Can information transmission be proven in a Rule 30 ECA?

(This is hopefully a clearer version of an earlier post of mine.) I have been spending lots of time on the open challenge of proving the aperiodicity of the central column of a rule 30 cellular ...
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### Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
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### Do all LCG-based PRNGs suffer from predictable patterns?

I needed to produce trivial (low-quality) random integers and remembered how simple linear congruential generators were to implement from school: Went to Wikipedia, found the first example which ...
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### Probability question: 2 people meet from other sides of the world, who find they have lived in the same 3 property numbers in the same order

To me the probability of this happening seems almost impossible... I'm British and a friend of mine is Chinese. I showed her a letter I had received which had my address on it. She spots my apartment ...
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### Two aspects of randomness

Consider a random sequence of integers 1, 4, 3, 8, 2, 5, 3, 8 ... The only sufficient condition for the sequence to be random is its unpredictability ie. probability of any number coming next ...
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### Using Ito calculus to prove that $\int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^t W_s d_s$

I am busy trying to teach myself some stochastic calculus and have come across a statement that I am trying to prove. How can I prove that \begin{align} \int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^...
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### Series of probabilities

Consider $Y_k=X_1+...+X_k$, where $X_k \in \mathbb{N}_0$ are i.i.d random variables and $E[X_1]<1$ $$\sum_{j=1}^{\infty} P(Y_j=j) \overset{!}{=}1$$ How can I verify that this equation is true or ...
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### Benford's law not working?

So I recently came across Benford's law and immediately tried to code it out but the answer I got was rather confusing. I think my code is correct I'm pretty sure it is but the result is just not the ...
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### Simple Random Walk with equal probability of +1 and -1.

You have 1D random Walk, with +1 of probability 0.5, and -1 of probability 0.5. What is the probability that you will reach +10 but never exceed -5? Attempt: The probability of getting +10 is easy, ...
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### upper bound for expected for special random variable

We throw three times independently perfect dice. For each random realisation of our throwing we have the random vector $$(x_{1}, x_2,x_3)$$ where $x_i\in\{1,2,3,4,5,6\}$ for $i=1,2,3.$ We now consider ...
### How do we show $P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B).$?
In some questions I have been going through here, I came across this inequality several times. $A$ and $B$ are RV. How do we show $$P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B)$$.