# Tagged Questions

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

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### Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 \end{... 0answers 54 views ### Particle in a box Say I have a point particle located at the center of a box and imagine that I give it a velocity v in some direction. It will bounce back and forth in different directions maintaining the same speed |... 1answer 10 views ### Extending Random Number Ranges I am provided with a random number \xi \in [0,1]. I check if a particular \xi_i \lt x is true and if so, I need to convert those random numbers within the range [0,x) into uniform range in [0,1]... 0answers 49 views ### Sequence of non-independent coin tosses Suppose that a sequence of coin tosses is due to be performed. Let p_i denote the probability that the ith coin toss lands on Heads and let X_i denote the corresponding indicator random variable ... 1answer 15 views ### Why does the average of a set of random numbers to the nth power approach 1/(n+1)? I got bored and started running a Java program to mess with stuff like this. I did a boatload of trials and averaged them all together, first for a random number squared. Quick pseudo-code: ... 1answer 26 views ### Invertible Uniform “PseudoRandom” Function Perhaps this is better suited to a cryptography stack exchange, but I thought I'd try in mathematics in case this question is more obvious than I initially thought. I'm looking for a function ~f:\{1,... 0answers 36 views ### Limiting distribution of infinite sum of weighted bernoulli? Let p_n be some fixed pulse, for example p_n =e^{-n^{2}} We have an infinite sum y = \sum_{n=-\infty}^{\infty} a_n p_{-n} where a_n are iid bernoulli random variables taking the values +/- \... 0answers 39 views ### Mean Value of a Random Process Consider a random process X(t) = Z(t)\sin(wt-Q). Here Q is a random variable taking values q in [-\pi/2,\pi/2] with PDF given by$$p_1^Q(q) = \frac{\cos(q)}{2}$$Z(t) is some random ... 1answer 51 views ### Is there a statistical measure of bitwise entropy? (Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I ... 0answers 39 views ### Practical example of superiority of randomized algorithm I'm looking for an example to show my students of an algorithm for which randomization of some kind leads to better performance on average. And I don't want that randomization to be of the Monte Carlo ... 0answers 26 views ### Covariance matrix of random vector of vectors I am a beginner in statistics and tried to research my question online without much success. Motivation: I am working on an undergraduate project in cosmology. My problem involves several scalar-... 0answers 15 views ### Using injective functions to prove randomness Suppose we have a random subset of \mathbb N. My intuitive notion of randomness represents the ability to encode arbitrary real number. Assume that A is a random subset according to the above ... 0answers 3 views ### How to calculate E[P_e(Y)] = \int^\infty_{-\infty}Q(\sqrt {2y})f_Y(y)dy X is Gaussian (0,1) random variable with CDF F_X(x)$$ Q(x) = 1-F_X(x) $$Y is an exponential 1/\gamma random variable with PDF$$ f_Y(y) = \begin{cases} (1/\gamma)e^{y/\gamma} &y\ge0 ...
I have the following problem: $n$ values $U_1, \ldots, U_n$ are chosen randomly and independently from the interval $[0,1]$. When choosing $U_i$, the probability that $U_i$ is smaller than a ...