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

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1answer
35 views

Geometrically distributed random variable

Let $X$ be a geometrically distributed random variable with parameter P. Compute the density of $X^2$. The density of the random variable $X$ is $$f_X (x) = p(1-p)^x \text{ for } x=0,1, \ldots,$$ ...
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2answers
21 views

{Maximum of independent exponential R.V} need help to understand

I have question about this question What I don't understand I do understand that P(2nd_max of {X1, X2....Xn}< t) = P(X1< t) P(X2< t).... P( Xn-1 < t). This will become the ...
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0answers
27 views

Maximum-Likelihood parameters for Bernoulli trials with decaying success probability

Say I have a sequence of independent bernoulli trials $X_0, X_1, \ldots, X_n$, with exponentially decaying success probability $\mathbb{P}(X_t=1)=r^{-t}$, for some unknown parameter $r$. How can I ...
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0answers
16 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
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2answers
46 views

Product & Ratio's of 2 Random Variables

I'm interested to know whether it's the case that for random variables $X$ and $Y$ whether or not the ratio of $X$ and $Y$ can be computed as the product of $X$ and $1/Y$. That is, Is $\frac{X}{Y} ...
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1answer
22 views

[Maginal distribution Calculation]problem: confusion about the Region of integration

I am confused about the marginal distribution of the joint probabilty function $f(x,y)$. Problem: the joint probabilty function $f(x,y)$ is $1/2$ when $x$ and $y$ are within a rotated square about ...
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0answers
29 views

Conditional probability of function of two RVs

I have two random variables, $X, Y$ and their joint pdf, $f_{XY}(x,y )$. I am able to find the marginal PDFs, $f_X(x)$ and $f_Y(y)$ using $f_X(x) = \int_{-\infty}^{\infty}f_{XY}(x,y)dy$ and similar ...
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0answers
22 views

Prove that the empirical measure is a measurable fucntion

This problem came from Schervish, Theory of Statistics, Sec. 1.4 Prob. 24. Suppose that $X_1, \ldots, X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. Prove ...
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2answers
33 views

Independent two random variables with uniform density

Let $X \sim U[-1,1]$ and $Y=X^2$. Show that $X$ and $Y$ aren't independent. Of course we have $F_X(x) = \frac{x+1}{2}$ and $F_Y(y) = \frac{\sqrt{y}+1}{2}$. But how can I find $$F_{XY}(x,y) = \Pr(X ...
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0answers
69 views

A nice sequence of random variables

Let $f:U\mapsto \mathbb{R}^k$ with $U\subset \mathbb{R}$ be a smooth injective function. Suppose that $\sqrt n(Y_n- Y)\to N(0,\Omega)$ in distribution with $Y=f(X)$. Define $X_n$ by ...
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3answers
135 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
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1answer
187 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 ...
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0answers
16 views

Multually exlcusive + indpedent random variable?

Two events can't be both multually exclusive and independent at the same time. However, I have one question regarding the following problem: "Let $X$ be the number of even tosses and $Y$ the number ...
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3answers
63 views

what does it mean by “$\min\{X,Y\}$” where $X$ and $Y$ are random variable?

I see this term often. find $P(\min\{X,Y\}< z) $ where $X$ and $Y$ are independent random variable . From the online source, I see people saying this as $P( \{X < z\} \cup \{Y< z\} ...
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2answers
46 views

Can this expectation be computed?

Suppose I have some function $f(\bullet)$ and a random variable $X$ with known distribution (let us say for example $X$ has a geometric distribution with mean $\displaystyle\frac{1}{1-\alpha}$). I ...
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2answers
73 views

Is a non-negative random variable with zero mean almost surely zero?

We have proven the following in class: If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements). Does it ...
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1answer
22 views

Condition on variable to make events independent

where, $$n_1,n_2,...,n_M \sim N\left(0,\frac{N_0}{2}\right) $$ how the condition on n_1 makes the events independent ? what is "n_1=n"
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0answers
8 views

function of Chi-square random variable

Assume that $\alpha$ is a Chi-squared random variable with $2$ degrees of freedom. what is the function $f(.)$, in order for the varible $\beta = f(\alpha)$ to be Chi-squared with $2N$ degrees of ...
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1answer
74 views

Roll a 6-sided fair die until a 6 appears. Let X = the number of 1's that are rolled. Find Var(X).

Let X = the number of 1's that are rolled. Find E[X] and Var(X). I can't seem to calculate Var(X). I've calculated E[X] = 1. I let R = the number of non-6 rolls, and I let Y = the number of rolls ...
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1answer
101 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
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0answers
30 views

Computing variance of a proportion

I had a question regarding this paper. In page 3, they show the way to estimate $\pi$ as $$ \pi = \frac{\lambda + p - 1}{2p - 1} $$ and then they proceed to compute the variance as $$ Var(\pi) = ...
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1answer
35 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
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0answers
52 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
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0answers
33 views

How do I replace part of a covariance matrix with a different covariance matrix?

Imagine that I have 100 random variables, numbered $x_1,\ldots,x_{100}$ that are generated with some unknown Gaussian generator $N(0,\Sigma)$. Using a large number of observations, I construct a ...
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1answer
87 views

Weak law of large numbers: counterexample for independent but not i.i.d. variables

Can someone please give me an example for sequence $\{X_n\} $ of independent random variables, such that $$ E[|X_n|]<5 $$ for each n, and such that the weak law of large numbers doesn't hold for it ...
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5answers
266 views

Sum of random numbers is divisible by $10$

Suppose that $15$ three-digit numbers have been randomly chosen and we are about to add them. What is the probability that the sum would be divisible by $10$? If there were only two or three random ...
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0answers
18 views

How to set covariance for Bivariate Logistic Distribution

This is the logistic distribution of single random variable (taken from Wikipedia). x = random variable mu = mean of all random variables s = variance. Now, I want to do a Bivariate logistic ...
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1answer
46 views

probability of the sum of i.i.d. RV with uniform distribution being $>x$

I am solving a question for applied stochastic processes homework and I am stuck on this part: Let $X_1,X_2,\cdots, X_n$ be independent identically distributed random variables with uniform ...
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1answer
51 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
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1answer
84 views

Random variables with joint density function

Let R be the rectangle $\ \{(x, y); 0 <= x <= 2, 0 <= y<= 1\} $, and let $\ f(x, y) = >k(x^2+ y^2)$ on R and zero elsewhere. (a) Find the value of k which makes f a joint ...
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1answer
37 views

Does higher variance imply a higher covariance?

Suppose I have three random variables A,B,C. if var(B) > var(C) does that mean cov(A,B) > cov(A,C)? Assuming neither is uncorrelated meaning cov(A,B) and cov(A,C) don't equal 0.
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1answer
27 views

A mapping that does not preserve convergence in distribution

I'm trying to come up with a map $H: \mathbb{R}^k\to \mathbb{R}^k$ and a sequence of random vectors $X_n\Rightarrow X$ in $\mathbb{R}^k$ for which $H(X_n)$ does not converge in distribution to $H(X)$. ...
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0answers
24 views

How can I visualize the sample space of a complicated random experiment?

A sample space $\Omega$ of a random experiment is the set of all possible outcomes $\omega$. When dealing with random variables (which, formally, is a map from $\Omega \mapsto R$), we usually operate ...
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1answer
42 views

How to deal with the following problem of correlated random variables?

I have the following information: $\left[ \begin{array}{l} {X_1}\\ \vdots \\ {X_K} \end{array} \right]$ are correlated random variables with (zero mean, unit variance) covariance matrix $\left( ...
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2answers
81 views

Maximum likelihood estimator?

I am looking at some questions from Mods 2010 and I can't figure this one out. I think my problem is technical... We have a sample (L1,R1), ...,(Ln,Rn) with Lj and Rj normally distributed independent ...
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2answers
46 views

Probability distribution in wireless channel?

Let suppose that I have a random variable $X_{mn}=\sqrt{\left(1/d_{mn}\right)^\alpha}\times h_{mn}$ wherr $d_{mn}$ is a random variable with uniform distribution and $h_{mn}$ is a random variable with ...
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2answers
38 views

Function of random variable

I have this question: Suppose P(X=0)=1/2 and P(X=8)=1/2. What's the value of E[Y] if Y=(X^2)? So I am having trouble understanding how to go about doing this ...
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4answers
108 views

Expected value of rolling dice until getting a $3$

I am having trouble with this question with regards to random variables and calculating expected values: Suppose I keep tossing a fair six-sided dice until I roll a $3$. Let $X$ be the number of ...
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2answers
41 views

Expected value of random variable

I have this question: What's the expected value of a random variable $X$ if $P(X=1)=1/3$, $P(X=2)=1/3$, and $P(X=6)=1/3$? I am very confused as to how I can work this problem out. I was thinking ...
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2answers
97 views

Conditional expectation of number of dice rolls

I've been given the following problem and I'd like to get a better understanding of how to solve it. A fair die is rolled successively. Let $X$ be the number of rolls needed to get a 6 Let $Y$ be the ...
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2answers
38 views

Discrete random variable with $f(x)=c(2x-1)$

I feel really stupid because I don't understand this example at the start of my textbook. The chapter is discrete random variables and it starts out with this example with no explanation. I understand ...
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1answer
39 views

Show: $X_n\xrightarrow{\mathcal{d}} X$, then $\mathbb{E}\lvert X\rvert\leqslant\liminf_{n\to\infty}\mathbb{E}\lvert X_n\rvert$

Let $X_n, X$ be random variables with $X_n\xrightarrow{d} X$. Show that then $$ \mathbb{E}\lvert X\rvert\leqslant\liminf_n \mathbb{E}\lvert X_n\rvert. $$ So let $X_n\xrightarrow{d} ...
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2answers
85 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
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1answer
42 views

Geometric random variable $X$, $Pr(X\ is\ even) =$?

Original Question: Toss an unfair coin until we get HEAD. Suppose the total number of tosses is a random variable $X$, and $Pr(HEAD) = p$. What is the probability that $X$ is even? Denote this event ...
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1answer
48 views

variance of a random variable

If $X_1, X_2 , ....., X_n$ iid $N(0,1)$ , and $S^2$ was defined as the population standard deviation we are to find the variance of $S^2$ I want to know the distribution in order to find the ...
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2answers
137 views

moment generating function technique

If $X$ was a random variable with a distribution $\mathrm{Normal} ( 0, 1 )$, using moment generating function technique we have to show that $Y= X^2$ has the Chi-square distribution with $1$ degree of ...
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2answers
26 views

existence of a RV with distribution given by a linear combination of other distributions

Question: Let $X$ and $Y$ be random variables defined on a $(\Omega,\mathfrak{F},\mathbb{P})$ probability space with distribution functions $F_X(t)$ and $F_Y(t)$, respectively. (a) Show that for any ...
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2answers
51 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
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0answers
57 views

Random variables plus a constant

Quick question: If a random variable $\mathcal{Z}$ with expected value 0 and $\sigma=10$ Volts, how do you take into account a constant voltage of 5 Volts added to it? Do I just add 5 to the density ...
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2answers
21 views

Variance of three possible outcomes

I am new to this kind of things so maybe you could help me get the reasoning. I have a continuum of outcomes on the interval $[0,1]$. Now, let us cut the interval into two pieces so that there are two ...