Questions about maps from a probability space to a measure space which are measurable.

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Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
4
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3answers
530 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
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1answer
15 views

Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
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1answer
27 views

Expected Value of Two Random Variables

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} ...
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1answer
26 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
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1answer
42 views

Convergence of expected values and integrability

I'm trying to prove a result for a homework assignment, and I got to a point that if the following result is true, then the result follows. Let $X_n$ be a sequence of positive random variables and ...
3
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1answer
36 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
4
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0answers
59 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
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1answer
31 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
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0answers
14 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
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0answers
9 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. Let its CDF be given by $F(\theta) := ...
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2answers
400 views

Probability of inequality between random variables

In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct. For three random variables $X,Y$, and $Z$ not necessarily ...
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1answer
24 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
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1answer
39 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
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1answer
37 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
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0answers
23 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
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2answers
47 views

what is the expected value of $x^TAx$? [on hold]

Assume $x\in \mathbb{R}^N$ is a random variable vector (like a noise sequence). You now want to calculate the following term: $E\{x^{T}Ax\}$, where $A$ is a constant matrix. How can this expression ...
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2answers
49 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...
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2answers
34 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [on hold]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
3
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2answers
70 views

If the variance is $0$ is it constant?

We know that the variance of a constant is $0$. Is the converse also true? Can we say that if the variance of some random variable is $0$ it is a constant?
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0answers
8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
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1answer
21 views

Uniform distribution and real values [on hold]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
3
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2answers
89 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
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2answers
66 views

Distribution of Summation of two discrete random variables

Here, $X$ and $Y$ are two non-negative independent discrete integer-valued random variable and the support set of $k_1$ and $k_2$ are $ \{ 2,3,...,7 \}$ and $ \{ 5,6,...,12 \} $ respectively. We ...
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2answers
208 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
4
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6answers
61 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) ...
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1answer
22 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + ...
0
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0answers
28 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
0
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1answer
290 views

partial differentiation of function of expectation of random variable

We have $E(U)=\int_0^B V f(V) dV + B \int_B^\infty f(V)dV$; Here $V$ is random variable. $E(U)$ stands for expectation of $U$. We have $Z=f(E(U))$ i.e. $Z$ is function of $E(U)$. Can we write ...
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0answers
21 views

Conditional expectation of discrete uniform random variables with one fixed

Came across a problem that I worked on sometime ago having the following structure: Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...
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1answer
28 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
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0answers
19 views

Probability distribution to failure [closed]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...
0
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1answer
32 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
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2answers
58 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
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2answers
22 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
0
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1answer
32 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
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2answers
56 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
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1answer
47 views

Existence of a global maximum of a function defined with the moment-generating function

Can someone give me an idea how to prove the following exercise? Let $Z$ be a real-valued random variable whose moment-generating function $m_Z$, with $m_Z(\gamma)= E\left[ \exp(\gamma Z) \right]$, ...
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1answer
52 views

For zero-mean r.v. $X$ with var. $\sigma^2$, want to show $E[e^{2X}]\leq e^{2\sigma^2}$.

Let $X$ have zero mean, $E[X]=0$ and finite variance $E[X^2]=\sigma^2<\infty$. I'm trying to show $$ E[e^{2X}] \leq e^{2\sigma^2}. $$ I started out with this related question, but I hadn't quite ...
2
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2answers
155 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
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1answer
27 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
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1answer
28 views

Convergence in proability does not imply convergence a.s. [duplicate]

I know convergence in probability does not imply convergence in measure. I would like to see some simple example. Do you have any ideas please?
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2answers
53 views

Bounding the expectation of a function of a zero-mean random variable

I have a random variable $X$ with mean zero, $E[X]=0$, and finite second moment, $E[X^2]=\sigma^2<\infty$. I'm wondering if it's possible to show the following bound: $$ E[(e^{X/2}-1)^2] \leq ...
2
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0answers
29 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
0
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0answers
14 views

Generating pseudo-random numbers around a distribution with an uncertain/chaotic mean

I originally asked this question on cross-validated, but apparently it is too mathematical a problem for that site. I want to simulate data collected by an instrument realistically. The problem is ...
0
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2answers
16 views

statistics question about two random variables

Two random variables $X$ and $Y$ have the following joint pdf: $$f(x, y) = \begin{cases} \frac35x(y + y^2) & \text{if }0<x<2\text{ and } 0 < y < 1\\ 0 & \text{otherwise} ...
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2answers
62 views

How to find distribution of order statistic

Let $X_i$ be iid random variables with common density $f$ and distribution $F$. Let $Y_k = X_{(k)}$ be the k-th order statistics (that is, $Y_1 = X_{(1)} = \min(X_i)$ etc.). Show that the joint ...
1
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0answers
23 views

mean and variance of this Gaussian random variable

I am trying to read through this paper - http://www.malcolmdshuster.com/Pub_2002c_J_scale_scan.pdf Equation 2(b)from the paper says [A] $\nu_k \equiv 2(B_k - b).\epsilon_k - |\epsilon_k|^2 $ where ...
1
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1answer
394 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
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1answer
86 views

How can I prove if $Y\leq X$ then $E[Y]\leq E[X]$?

If $Y\leq X$ always holds, then $E[Y]\leq E[X]$. How can I prove this (formally)? Also, can the equality happen if we know that $Y=X$ does not always hold? (i.e. $X$ and $Y$ are not exactly the same)