# Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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### The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
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### For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
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### Possibility of analogy of the Bayes theorem for expectations

I recently found a scenario where I wanted to find the relation of $E[X|Y]$ and $E[Y|X]$ for $X,Y$ two random variables. For probabilities and densities we have the Bayes theorem which is well ...
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Let $X_1, X_2, \dots, X_n$ -- i.i.d. sample. Suppose we have two models: $$M_0: X_1, X_2, \dots, X_n \sim N(0, 1) \\ M_1: X_1, X_2, \dots, X_n \sim N(\theta, 1)$$ So we have two hypothesis: $H_0: ... 0answers 30 views ### Variation on the classic ABRACADABRA problem Suppose letters are chosen randomly from the alphabet, one at a time. It is well known that the expected time until ABRACADABRA is spelled out is$26^{11}+26^{4}+26$. The proof uses discrete time ... 0answers 23 views ### Distribution that describes # of memoryless events in an interval but the mean is not constant. Context: I'm taking a stochastic processes class right now and we got a bit into queueing theory. In all the queue's we've considered, arrivals follow a poisson process. This seems unrealistic in some ... 0answers 33 views ### Are Ito Integrals adapted to the Brownian Motion Filtration Give a probability space$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_t, P)$, we could define a 1-dim Brownian motion$W_t$adapted to$\{\mathcal{F}_t\}_t$with its own filtration$\mathcal{F}_t^W$. For ... 0answers 28 views ### Convergence in distribution implies convergence of sequence Let$(X_n)$be a sequence of random variables. Let$(a_n), (b_n)$be sequences of real number. Assume that there exists random variables$X, Y$such that $$X_n \rightarrow^{d} X, a_nX_n + b_n ... 0answers 18 views ### If B_t - B_s, \ 0\leq s < t, is normally distributed, there are constants C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n I am working on the following problem: Show that if B_t - B_s, 0 \leq s < t, is normally distributed with mean zero and variance t-s, then for each positive integer n there is a positive ... 0answers 27 views ### Generating continuous random variables from a set of Bernoullis Given a set of Bernoulli(p_i) variables with each having its own arbitrary p_i, is there an efficient way to generate continuous random variables sampled from an arbitrary distriubution? To ... 0answers 16 views ### Fourier transform of a probability measure, and fourier transform of density I have defined, for a probability measure \eta we have the fourier transform as \hat{\eta} = \int e^{itx} \ d\eta(x), and for a function h: \mathbb{R} \to \mathbb{R} we have that the fourier ... 0answers 9 views ### Natural \sigma-algebra on the co-domain of a stochastic flow Say that we have a stochastic local flow X. This is a map that sends \omega into a collection of maps \phi_{s,t} on s,t\in \mathbb{R} which form a local flow. I don't understand how X ... 0answers 42 views ### Finding the Cramer-Rao Lower Bound Given the probability density function$$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$and$0$otherwise, find the Cramer-Rao Lower Bound ... 0answers 17 views ### Formal definition of expectation question I have expectation of a random variable on a probability space$(\Omega,\Sigma,P)$defined as: If X is lebesgue integrable with respect to$P$then$EX = \int_\Omega X \ dP$. What I don't understand ... 0answers 33 views ### Product of expectations is a martingale Consider a probability space$(\Omega, \mathcal{F}, P)$and random variables$X_0, X_1, \ldots , X_n$adapted to the filtration$\{\mathcal{F}_t\}_{t\geq0}$. Assume furthermore that each$X_n$is ... 0answers 32 views ### A question on conditional expectation leading to zero covariance and vice versa In my probability class I was tackled with this seemingly weird question involving conditional expectation: Let X,Y be two random variables (it is not mentioned whether or not they are discrete or ... 0answers 27 views ### Transform Markov chain that doesn't have stationary transition probabilities to one that does? This question concerns Exercise 7.3 in Walsh's Knowing the Odds. A Markov chain is defined as having stationary transition probabilities if for all$i, j, n$we have$P(X_{n+1} = j \mid X_n=i) = ...
I have a problem, to whose answer I want an intuitive explanation. Let $X_1,\cdots,X_5$ be independent continuous random variables having a common distribution function $F$ and density $f$. Let \$I= ...