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

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3
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1answer
43 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
1
vote
1answer
21 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
5
votes
1answer
186 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
2
votes
1answer
43 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
3
votes
2answers
172 views

Variance of a function of independent random variables

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
1
vote
2answers
50 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
0
votes
0answers
56 views

Random Variables Proof Help [on hold]

I have two random variables $X$ and $Y$. Each of them having the conditions of Markov´s Inequality. So $X$ and $Y$ take on only nonnegative values, then for any value $a > 0$ $$P\{X \ge a\} \le ...
3
votes
1answer
50 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
1
vote
0answers
22 views

Proving asymptotic normality

Suppose I have a sequence of independent random variables. Then how do I prove $S_n\sim\mathrm{AN}(0,n)$. Lyapunov or Lindeberg's CLT not working since $V(S_n) \neq n$ and the Characteristic function ...
0
votes
0answers
20 views

For a sequence of random variables with bounded probability density function, can their joint pdf be unbounded?

For $d\geq2$, let $X_{i}=\left\{Y_{i-1},Y_{i-2},...,Y_{i-d} \right\}$, and assume the sequence $\left\{X_i \right\}$ is strictly stationary. Let $f_{j}(x_{0},x_j)$ denote the joint density of ...
0
votes
0answers
26 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
0
votes
1answer
17 views

Sum of binomial distributed random variables

Let $X \sim Bi(n,p), Y \sim Bi(m,p)$. “Visual arguments” suggest that $X+Y \sim Bi(m+n,p)$. However, I am unable to prove that. Using the definition I can reduce the problem to $$\sum_{i=0}^k ...
0
votes
1answer
40 views

probability of getting 5 calls in 5 minutes

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. 1 a. Compute the probability of receiving three calls in a 5-minute interval of time. b. Compute the ...
-1
votes
1answer
31 views

probability that 4 out of 20 investors have exchange trade [closed]

1 out of 4 investors have exchange traded fund in their portfolio. consider a sample of 20, what is the probability that exactly 4 investors have exchange traded fund.
0
votes
1answer
30 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
0
votes
2answers
32 views

On the definition of a random variables

Let $(O,F,P)$ be a probability space. That is $O$ is a set, $F$ is a $\sigma$-algebra of subsets of $O$ and $P$ is a probability measure. Consider a function $f:O\to\mathbb R$. Would we call $f$ a ...
1
vote
0answers
26 views

A Property of Conditional Expectation [closed]

Suppose $X$, $Y$, $Z$ are three random variables. $\sigma (X,Y)$ is independent of $\sigma(Z)$. $f(Y,Z)$ is a measurable function. Then, does the following identity: $E[X\,|\, f(Y,Z),Y]=E[X\,|\,Y]$ ...
0
votes
1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
1
vote
5answers
2k views

The sum of n independent normal random variables.

How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
0
votes
0answers
50 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
1
vote
1answer
294 views

Maximum entropy joint distribution from marginals?

How does one find the maximum entropy joint distribution of two random variables X and Y given their marginal probability mass functions? I know: I have the marginals, meaning p(x) and p(y) are ...
0
votes
0answers
23 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
0
votes
2answers
37 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, ...
2
votes
1answer
106 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
0
votes
1answer
45 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 ...
3
votes
1answer
70 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 ...
0
votes
0answers
8 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] ...
-1
votes
1answer
23 views

Are these variables correlated [closed]

Given that $a,b,c,d,p,q$ are independent random variables, are $X=ab-cd$ and $Y = cp - aq$ correlated? If not, are they independent?
1
vote
3answers
74 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 ...
0
votes
1answer
12 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 ...
0
votes
0answers
18 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 ...
1
vote
0answers
15 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 ...
0
votes
0answers
17 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 ...
1
vote
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
vote
1answer
26 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) = ...
0
votes
1answer
53 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 ...
1
vote
2answers
42 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.
0
votes
2answers
398 views

What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Ok, so I think I have a working solution to this problem. Heres how I would solve it: so you look at a 6 appearing as a success and everything else as a failure. So from here you can you use the ...
-1
votes
0answers
20 views

Almost surely convergence with stationary random vectors

I dont seem to be able to incorporate the stationarity condition into any of limit theorems I know. I cannot see how the Birkhoff almost everywhere ergodic theorem could be used as I cannot see how ...
0
votes
2answers
95 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and ...
2
votes
1answer
70 views

Discovering the joint distribution for two dependent random variables?

Suppose I have three continuous random variables $X_1$, $X_2$, and $Y$, where $Y = X_1+X_2$, and $X_1$ and $X_2$ are dependent. If I know the probability distributions separately for $X_1$, $X_2$, ...
1
vote
1answer
246 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
1
vote
1answer
84 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
vote
1answer
54 views

finding the limits of integration for joint probability

I have three variables $x_1$, $x_2$ and $x_3$. Their joint dist. is $f(x_1,x_2,x_3)= \exp(-x_1-x_3)$, where limits of $x_3 = 0$ to $\infty$, $x_2 = x_3$ to $\infty$ and $x_1 = x_2-x_3$ to $\infty$. ...
5
votes
2answers
270 views

Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As ...
1
vote
1answer
21 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) = ...
0
votes
0answers
21 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
14 views

Could you explain intuitively the mean & variance of Chi-square random variable?

$X=X_1^2 + X_2^2 + ... +X_n^2$ , where each $X_i , i=1,...,n$ is i.i.d and and Gaussian distributed with mean $0$ and Varaiance $\sigma^2$. Why is mean of $X$ is $n\sigma^2$ and variance is ...
1
vote
1answer
21 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 ...