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

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2
votes
1answer
40 views

Law of large numbers, problem

I have a specific problem to solve using strong law of large numbers. Let $X_k$ be independent uniform random variables on interval $(0,k)$. Let $Y_n ={1 \over n^2}\sum\limits_{k=1}^n {X_k^3 \over ...
2
votes
1answer
40 views

Proving a certain limit for uniformly integrable random variables

There is an interesting problem that has been resisting my efforts for a while. Assume that $\{X_n: n = 1, 2, \ldots\} $ is a sequence of uniformly integrable random variables. I would like to show ...
0
votes
1answer
34 views

Why is this a martingale.

So I'm looking at this page http://notesofastatisticswatcher.wordpress.com/2012/01/05/a-martingale-that-tends-to-latex-infty-with-probability-1/ where they have this martingale that goes to $-\infty$ ...
3
votes
1answer
52 views

Bound the variance of the product of two random varables.

For two random variables $X$ and $Y$ show that the following inequality holds $$\mathrm{Var}(XY)\leq 2\|Y\|_{\infty}^{2}\mathrm{Var}(X)+2\|X\|_{\infty}^{2}\mathrm{Var}(Y).$$ Well first I tried to ...
2
votes
1answer
46 views

An independent squence of functions that are uniform on $[0,1]$

Suppose that $X$ is uniform in $[0,1]$. Find an infinite sequence of functions $f_{i}$ so that all $f_{i}(X)$ are independent and uniform $[0,1]$. um I'm not really sure how to do this. I'm thinking ...
1
vote
1answer
10 views

Fisher exact text and connection between Binomial and Hypergeometric distributions.

My textbook shows the connection between binomial and hypergeometric using the fisher exact test.."Assuming the null hypothesis and letting p=p1=p2, we have $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$, ...
0
votes
1answer
32 views

How should I approach this Conditional Probability Problem?

Can anyone give a hint on how to begin this problem? Suppose $Y = X^2 + W$ where $W$ is Gaussian $N(0, 1)$ noise. Then derive an expression for $P(Y\mid X)$. I know about Bayes' Rule but I'm not ...
3
votes
0answers
21 views

Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
0
votes
0answers
8 views

Generate random numbers with beta distribution from uniform distribution

How can I generate a series of random numbers with beta distribution from random numbers with uniform distribution? I am aware that using inverse transformation method is at least very difficult or ...
0
votes
0answers
12 views

How can I find the nonlinear and linear MS estimates of y in terms of x and the resulting MS errors?

If $y=x^3$, find the nonlinear and linear MS estimates of $y$ in terms of $x$ and the resulting MS errors? This is what I got for the nonlinear MS estimation: Since $e=E\{[y-C(x)]^2\}$, $C(x)=x^3$ and ...
0
votes
1answer
21 views

Beta-binomial random number generator

Could someone help me find a random number generator from a Beta-Binomial distribution in MATLAB, R or SAS? Thank you!
3
votes
1answer
26 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
2
votes
2answers
76 views

Expectation Random Variables

Say $X$ to be uniformly distributed from $[0,1]$. Say $k_1$ and $k_2$ to be two non negative constants (that is, they take values from $[0,+inft]$. I want to compute the expectation of the following ...
1
vote
2answers
19 views

iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
0
votes
2answers
23 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because ...
1
vote
1answer
39 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
1
vote
1answer
53 views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
0
votes
0answers
44 views

Dynamic programming for optimal maximum and optimal minimum

We have a sequence of $a_i$ and a choosing rule that is take the first number $x_t\ge a_t$. The definition is = $$ min\{ t|t \in \{ 1,2,\cdots,n\}\,\,,\,\, x_t\ge a_t\}$$ The sequence $a_i$ is ...
0
votes
0answers
27 views

Independence of Bernoulli r.v. and product

Let $X_1,X_2$ be independent random variables each assuming only the values $+1$ and $-1$ with probability $1/2$. Are $X_1,X_2,X_1X_2$ pairwise independent ? Are $X_1,X_2,X_1X_2$ an ...
1
vote
0answers
15 views

question about exponential distribution or exponential random variables

Consider a post office that is run by two clerks. Suppose that when Mr. Anderson enters the system he discovers that Mr. Smith is being served by clerk 1 and Mr. Brown by clerk 2. Suppose also that ...
0
votes
2answers
83 views

Expected value of the sum of the two largest values from a Uniform parent

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?
-1
votes
2answers
69 views

Show that Y=aX+b is an random variable. [closed]

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
1
vote
0answers
11 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
0
votes
0answers
21 views

Properties of Identically Distributed RVs.

I've a little doubt in part (iii) of the question posted above First I wrote the PMF of Z \begin{vmatrix} Z = X+Y & -2 & -1& 0 & 1 & 2\\ P(Z=z) & .09 & 0.24 & 0.34 ...
0
votes
1answer
26 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
1
vote
0answers
26 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
0
votes
0answers
21 views

A random variable with distribution continuous on a parameter: Is a continuous function of such random variable continuous in the parameter?

Let $(X_n(\lambda))_{n\in\mathbb{N}}$ be a sequence of i.i.d. real continuous random variables (with density function) and assume that $P(X_n(\lambda)\le x)$ is continuous in $\lambda$. Consider the ...
0
votes
1answer
21 views

If $X$, $Y$ are IID Gaussians, why is $U= X + 2Y$ independent from $V= Y-2Y$?

It seems to me that if $U$ and $V$ are made up of the same variables $X$ and $Y$, then they should be related in some way. I know that the covariance is 0, but dependent variables can in some cases ...
0
votes
3answers
45 views

Formal proof that X and X squared random variables are dependent.

Intuitively I know that any $X$ and $Y = X^2$ random variables are not independent, but I can't come up with a formal proof. In the case I'm most interested in, $X$ is uniformly distributed on ...
0
votes
1answer
29 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
0
votes
1answer
14 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
2
votes
1answer
32 views

Tail probability of a max of iid

If $X_{i}$ are iid random variables with $X_{i}>0$ and $\mathbb{P}(X_{i}>t)\sim t^{-\alpha}$ as $t\to \infty$. Then my question is: Is it also true that $\mathbb{P}(\max_{1,\dots n} ...
0
votes
0answers
19 views

what will be the PDF of magnitude and phase of this random variable?

i have a random variable as shown in the figure and i tried to find the PDF of the magnitude and phase of this random variable using central limit theory as i mentioned , i know that if we have ...
0
votes
1answer
27 views

Proof: $\sum\limits_{n=1}^\infty \mathbb E(|X_n|)< \infty \Rightarrow \sum\limits_{n=1}^\infty X_n$ converges almost surely

I was reading this as a Lemma, however my book doesn't provide proof of it: Let $X_1,X_2,...$ be a sequence of random variables, then the expression in the title is true. I'm trying to ...
0
votes
0answers
13 views

Conditional random variable confusion

I am trying to understand a step of reason. It goes like this $$E[E(Y^2|X)]=E[\mathrm {var}(Y|X)+(E(Y|X))^2].$$ But shouldn't the step be $$E[E(Y|X)^2]=E[\mathrm {var}(Y|X)+(E(Y|X))^2].$$
3
votes
1answer
47 views

Where do I go wrong?

Suppose $X,Y$ are independent Uniform$(0,1)$ random variables. Find the probability $P(Y\geq X\mid Y\geq\dfrac{1}{2})$. Please note that I know the correct answer and that I have arrived at the ...
3
votes
1answer
18 views

Expectation of uniform random variable knowing sum of $n$ identical uniform variables.

Let $X_1, ..., X_n$ be independent and identically distributed random variables on $[0,1]$. Find: $$ \mathbb{E}[X_1|X_1 + ... + X_n = x] $$
1
vote
1answer
24 views

Rewriting Gaussian r.v. $Z$ as sum of two independent Gaussian r.v.

Suppose, $Z$ is Gaussian r.v. assume that it has mean 0 an variance 1. My question is can $Z$ be rewritten as \begin{align*} Z=\rho Z_1+(1-\rho)Z_2 \end{align*} where $Z_1$ and $Z_2$ are independent ...
0
votes
1answer
11 views

relationship between two normally distributed variables

Say I have two normally distributed independent random variables (X1 and X2) with the same variance but different means. How would I calculate P(X1 > X2)?
1
vote
1answer
35 views

Comparing Sample Mean and a Random Variable

Let $X_{(i)} = ( i = 1,2, \ldots, n+1)$ be a random sample of size $n+1$ that is produced from a normal population. Let $M$ be the sample mean of the first $n$ random variables in this random sample. ...
1
vote
1answer
24 views

Expected score in marksmanship competition.

Problem: Marksmanship competition at a certain level requires each contestant to take ten shots with each of two different handguns. Final scores are computed by taking a weighted average of 4 times ...
1
vote
1answer
38 views

Show $P(X|Z_1,…,Z_n,Y)\not = P(X|Z_1,…,Z_n) \Leftrightarrow P(Y|Z_1,…,Z_n,X)\not = P(Y|Z_1,…Z_n)$

If we have two random variables $X,Y$ and a set of random variables $\{Z_1,...,Z_n\}$, are there any common proofs of the result in the title? Which theorems does this follow after?
0
votes
1answer
39 views

Further explanation regarding calculation of E[X^2]

I was reading over the following evaluation of $ E[X^2] $ on the following pdf: http://crab.rutgers.edu/~guyk/dmlec/lectures/lec15/l15.pdf. This part was especially confusing for me: ...
-2
votes
0answers
11 views

Find $P(0.5\le X\le 2, 0\le Y\le 1)$ given $X$, $Y$ continuous random variables and pdf

$X$ and $Y$ are continuous random variables with joint pdf; $$ f(x,y) = \begin{cases} \dfrac{6}{11}x(x^2 +y^2) & 0\le x\le 1; 0\le y\le 2\\[2ex] 0 & \text{otherwise} ...
0
votes
1answer
34 views

Expectation of the derivative of a random process

Let's have a Random Process $Y(t) = X(t) + 0.3 X'(t)$ Mean of $X(t) = 5t$ Question : Find the mean function of $Y(t)$ What I did : $E(Y) = E(X) + 0.3\cdot E(X')$ ? I don't know if I have ...
-2
votes
2answers
19 views

Variance of $2X_1 +X_2+3X_3$ with $X_i \sim \operatorname{Poisson}(i x \lambda)$

$X_1, X_2, X_3$ are independent random variables such that $X_i \sim \operatorname{Poisson}(i x \lambda)$, $i=1,2,3$. What is the variance of $2X_1 + X_2 +3X_3$? I know how to find ...
0
votes
1answer
28 views

Resource for functions of random variable problems

Let $X_{1}$ and $X_{2}$ be two random variables with jpdf: $f(X_{1}, X_{2}) = 4X_{1}X_{2};$ for $0<X_{1}<1, 0<X_{2}<1$ Find the probability distribution of $Y_{1} = X_{1}^{2}$ and ...
0
votes
0answers
13 views

how can I Find a 95% credible interval for p using the Bayesian method with the uniform distribution as a prior for p?

When I have a RV X~Geom(p): $x\ Frequency\\ 1 7459\\2 1930\\ 3\ 463\\ 4\ 117\\ 5\ 22\\ 6\ 6\\ 7\ 2\\ 9\ 1$ This is what I am trying to do: Since p is a probability, I say that $ p\sim U[0,1]$ An ...
0
votes
1answer
25 views

Distribution for random variable Z = Y1 - Y2

This was one of the interview questions. I did not know the answer. Question : Let Y1 and Y2 be two independent random variables where Y1 follows Normalpdf[x, -2, 5] distribution and Y2 follows ...
0
votes
1answer
12 views

Finding density functions from conditional distribution

I'm currently taking a statistics course, but I'm having trouble with a specific concept, and hope this is a good place to ask. Essentially, for random variables $y_{1},y_{2}$, how do you get from ...