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

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45 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
0
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0answers
12 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
3
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1answer
72 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
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1answer
15 views

quick question on measurability of random variables and what becoming a deterministic function means.

we stated a theorem in class: if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y. This is fine. The Professor sometimes states that X ...
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0answers
108 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
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0answers
18 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
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1answer
73 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
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0answers
29 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
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3answers
83 views

Finding expected value??

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the ...
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0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
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1answer
29 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
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1answer
58 views

Prove that E(X) exists if and only if E(|X|) exists.

I found this theorem in a book, but there is no proof there: If X is a random variable, then Prove that E(X) exists if and only if E(|X|) exists. where $E(X)$ is the expected value of $X$ I know ...
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2answers
48 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
1
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1answer
111 views

Generating random variates in Excel

I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't ...
1
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1answer
22 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
1
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0answers
34 views

Independent of random variables.

When reading Shiryaev's Probability. In the chapter 1, section 4. problem 11: Show that the random variables $\xi_1,\cdots,\xi_n$ are independent if and only if ...
1
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1answer
25 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
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2answers
23 views

Expected Value with Parameter p

The random variable X has the following probability distribution: P[X=-1]= (1-p)/2 P[X=0]= 1/2 P[X=1]= p/2 The parameter p satisfies the inequality $0 < p < 1$. Find the expected value and ...
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4answers
40 views

Generate random numbers in a random fashion

How can I generate 9 random numbers between 1 to 9,without repetition, one after another. Its like: Lets assume that the first random number generated is 4, then the next random number has to be in ...
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1answer
85 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
1
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2answers
123 views

Expectation of random variables ratio

Let $X_1, X_2, \dots, X_n$ be $n$ positive iid random variables. Then show that $$E\left(\frac{\sum_{j=1}^k X_j}{\sum_{i=1}^{n} X_i}\right) = \frac{k}{n}.$$ Because of the linearlity of the ...
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1answer
103 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
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1answer
18 views

restricting range of values that random variable can take

I have a random variable $Y$ such that $Y=X+\epsilon$ where $X$ is not random variable and takes values in $(0,1)$. $\epsilon$ is random variable with $E(\epsilon)=0; Var(\epsilon)=\sigma^2$. What ...
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0answers
31 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
6
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1answer
77 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
7
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0answers
88 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
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1answer
25 views

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum X_i$, sum of independent Bernoulli trials with different $\theta_i$. Find $\operatorname{Var}(Y)$

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum_{i=1}^n X_i$, sum of independent Bernoulli trials with different $\theta_i$. So this is something like we have a collection of $n$ possibly ...
1
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1answer
36 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
1
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1answer
46 views

Numerical CDF estimation for complicated random variable

Given a combination $U$ of several random variables $X,Y,Z...$ with known distributions, what is an efficient numerical algorithm to estimate PDF or CDF of $U$, if its CDF, PDF, characteristic ...
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1answer
40 views

Show that $\Pr(S_N\in A\mid N=n)=\Pr(S_n\in A)$

Let $X_1,.\ldots,X_n$ be i.i.d. random variables and $N$ be a positive integer-valued random variable, which is independent from the sequence. If $S_n=\displaystyle\sum\limits_{i=1}^{n} X_i$, then ...
0
votes
1answer
19 views

How to find the LimInf of a sequence of RV's

I have an indipendent sequence of Rv's $Y_n$ with law that eventually is: $\mathbb{P}(Y_n<a) = \begin{cases} 0 & \mbox{if}\ a \leq 0 \\ 1 - \frac{1}{n^{2\gamma}a^2}& \mbox{if}\ a ...
0
votes
2answers
53 views

X and Y are independent random variables and their distributions are..

X and Y are independent random variables and their distributions are.. $P(X=1) = 0.1 $ $P(X=2) = 0.2$ $P(X=3) = 0.3 $ $P(X=4) = 0.4 $ $P(Y=4) = 0.4 $ $P(Y=2) = 0.3$ $P(Y=3) = 0.2 $ $P(Y=4) = 0.1$ I ...
2
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0answers
26 views

Density of the $k^{th}$ smallest of $X_1,X_2,…,X_n$

Show that if $(X_1,X_2,...,X_n)$ are i.i.d. with common density $f$ and distribution function $F$, then $X_{(k)}$ has density $$f_{(k)}=k\binom{n}{k}f(y)(1-F(y))^{n-k}F(y)^{k-1}$$ where ...
0
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1answer
31 views

Evaluating an expectation of the supremum of collection of random variables

I know that $\mathbb{E}(sup_n|X_n|)=\infty$ if $X_n=\frac{2^n}n\cdot\mathbf 1_{(1/2^{n+1},1/2^n)}$. However I am not sure how this can be evaluated explicitly. The probability space is $Ω=[0,1]$ ...
0
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1answer
50 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
0
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1answer
26 views

$X $ and $Y$ are continuous $RVs$, such that$ f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$

X and Y are continuous RVs, such that $f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$ I'm trying to find $P(x<1/2,y>1/2)$. So i'm integrating from $\dfrac{1}{2}$ to $1$ for $y$ ...
0
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1answer
31 views

Covariance of random variables with identical distribution.

Let $X_1,...,X_n$ be random variables with identical distribution, and for all $i=1,...,n$ $\mathrm{Var}(X_i)$ exist. 1. Show that the covariance between each two random variables exist. 2. Show that ...
1
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1answer
58 views

First hitting time expectation and Markov property

Let $H_A$ be the first hitting time, such that $H_A\geqslant1$, so we have $X_0=i\notin A$. All texts I looked at, state without any further justification that $$ \mathbb E(H_A\mid X_1=j, ...
2
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0answers
1k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
0
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1answer
122 views

Find the PDF of Y given Y=X(2-X) and X's PDF

Suppose that the continuous random variable $X$ has probability density function $f_X(x)=\begin{cases}\frac{1}{2}x & \text{if } 0<x<2\\0&\text{otherwise}\end{cases}$ Let $Y=X(2-X)$. ...
0
votes
1answer
187 views

If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$ X = \sum_j^r l^j Y^j, $$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
3
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1answer
44 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
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2answers
184 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ ...
0
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1answer
74 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
0
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1answer
22 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
2
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0answers
23 views

does the hilbert space construction of random variables allow for infinite variance?

I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner ...
0
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1answer
23 views

Uniform Spinner is spun twice..

A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1]. I'm trying to answer the question what is the joint pdf for V and W. I know that I have to ...
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2answers
407 views

Expected Value of Intersection of two Binomial Random Variables

Ok the problem is as follows: (I am currently studying for my first actuary exam so this isn't a specific hw question! Just trying to figure it out!) A and B will take the same 10-question exam. ...
2
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2answers
62 views

Find the distribution of random variable $XY+X+Y+1$

X and Y are iid with density $f(x)=\frac{1}{(1+x)^2}I_{(0,\infty)}$. Find $P(Z\le z)$ where $Z=XY+X+Y+1$ my effort: $P(Z\le z)=P((x+1)(y+1)\le z)=P(x\le ...
1
vote
1answer
53 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...