# Questions tagged [random]

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

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### Spectrum of sum of (weighted) random matrices

Coming from statistical physics, I am interested in the spectrum of the following sum \begin{equation} \sum_{n=1}^m c_n X_n, \end{equation} where $c_n$ are non-random real numbers and $X_n$ are ...
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### Complete randomness/disorder and determinism

Wikipedia page about randomness says that "complete disorder" and "true randomness" are impossible according to Ramsey Theory and Cristian S. Calude. I don't understand it. https://...
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### Unbiased estimator given pdf

Let $X_1,X_2,…,X_n$ be independent samples from a distribution with pdf $f_X(x)=\frac{1}{θ^2}xe^{-\frac{x}{θ}}$ $(X≥0)$. Which of the following is an unbiased estimator for $θ$ ? I'm trying to solve ...
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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 ...
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### Long string of same rolls in a random sequence of dice throws

I'm not a mathematician and my knowledge of probability and statistics is limited. I came across an interesting problem, while discussing randomness in games, and honestly I don't even know where to ...
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### Are the last digits of a uniformly distributed random sequence going to always appear at the same frequency approximately?

I noticed that one can sample a sequence of uniformly distributed integers (user IDs in my case) algorithmically in sizes of 10%, 20%.. etc. by only choosing those numbers whose last digits equals one ...
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### Does intersection mean set theoretic point of view for independent events?

It is evident that when two events $A$ and $B$ are independent, then $P(A\cap B)=P(A)P(B)$ A good example is, "Tossing a coin and rolling a die". What is the probability that heads occurs on ...
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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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### Random Shuffle of Groups

Let's suppose that we have 54 peoples and we arrange them into 9 groups of equal size, so this means that each group will have 6 persons in it. I want to find a procedure, such that the groups are ...
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### Normally Distributed and Covariance is zero [duplicate]

Let $X \sim N(0,1)$, and let $P(Y=1)=P(Y=-1)=\frac{1}{2}$. Assume $X$ and $Y$ are independent, and let $Z = XY$. Prove that $Z \sim N(0,1)$ and $Cov(X,Z)=0$ For proving $Z$ is a standard normal ...
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### Distribution of random numbers with fixed sum

I have access to a black box function $f$ that returns 4 random integers $n_1$, $n_2$, $n_3$, $n_4$ with $4 \le n_i \le 13$ and $\sum_i n_i = 25$. Experimentally, I can see that $n_1$, $n_2$, $n_3$ ...
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### Normalising Low Duty-Cycle Random Data

I'm an engineer by training and developing a quantum random number generator for high-security cryptographic applications. I'm using single-photon arrival times as my source of entropy for the device. ...
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### A Probability Question From HW - Why a Random Variable Has Two Distribution?

Assume we have a random number generator that can generate a random number uniformly distributed in the range $[0, 1]$. If now we want to use this generator to generate a random number X that has a ...
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### Randomly guessing answers on a randomly generated test

My question stems from answering questions (on a test) whose answer order was randomly chosen (not by a human, by a computer). I’m skeptical of something I just read: Your overall score will improve ...
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### Product distribution of two uniform distributions which are centered around 1

Consider the product distribution $Z = X_1\cdot X_2$ for  \begin{aligned} X_1 &\sim \textrm{Uniform}[1 - a, 1 + a] \quad, \quad 0 < a < 1 \\ X_2 &\sim \textrm{Uniform}[1 - b, 1 + b]...
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### Family vs. Child when a girl is chosen, what is the probability that the second child is a girl, textbook clarification?

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I was able to understand the difference between selecting a child and a family, for ...
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### Decoupling two-dimensional simple symmetric random walk

It is an observation that the two-dimensional simple symmetric random walk can be broken into two independent one-dimensional random walks : that is, let $S_n$ be $2$D SSRW on $\mathbb Z^2$. The ...
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### What is the optimal strategy of guessing a number where closest without going over wins?

When a group of people need to decide a winner or leader between them, one approach would be that a random hidden integer is chosen with uniform distribution on $\{0, 1, ..., n\}$ and all $p$ ...