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

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2 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 ...
2
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1answer
11 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
45 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 ...
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2answers
12 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 ...
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2answers
17 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 ...
0
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1answer
17 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 ...
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1answer
21 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 ...
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0answers
19 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 ...
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0answers
22 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 ...
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0answers
12 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 ...
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2answers
57 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)))?
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2answers
62 views

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

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 ...
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0answers
10 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 ...
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0answers
18 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 ...
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1answer
23 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 ...
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0answers
23 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)) ...
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0answers
20 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 ...
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1answer
19 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 ...
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3answers
42 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 ...
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1answer
24 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
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1answer
11 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
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1answer
28 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} ...
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0answers
17 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 ...
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0answers
7 views

Integration of a Random Process

Let {$X_t$} be a strictly stationary random process defined for all time $t$; in particular, the distribution (PDF) at any time is the same. Let $Y$ be a random variable defined by $Y = \int_0^5X_t ...
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1answer
26 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
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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
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1answer
43 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: $$ \mathbb{E}[X_1|X_1 + ... + X_n = x] $$
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1answer
21 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 ...
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1answer
10 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)?
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1answer
33 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. ...
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0answers
13 views

Finding the infinitesimal generator of a M/M/2 queue [closed]

I have a M/M/2 queue with a total population of 5. The arrival times are independent exponential random variables with mean of $\lambda$ and the service times have a mean of $\mu$. The initial number ...
1
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1answer
23 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
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1answer
36 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?
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1answer
38 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: ...
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0answers
9 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
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1answer
30 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 ...
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2answers
16 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 ...
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1answer
26 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 ...
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0answers
10 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
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1answer
24 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
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1answer
11 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 ...
0
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1answer
26 views

Given 2 Random Variables, Please fill out the table

I am working on a problem and I have no clue where to start. I'm not sure what It is asking, or where to start. If you guys could give me the steps to take, show me what concepts are used, or a ...
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0answers
26 views

What is $E[\cos X]$ where $X$ is lognormal?

I was asked in an interview to compute $E[\cos X]$ where $X$ is lognormal. I tried using lognormal's characteristic function (Taylor series representation, which is divergent) and $\cos ...
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0answers
18 views

Measurability and random variables

Let $(\Omega, \mathcal{B})$ be a measurable space and $X$ a r.v. taking values in $\mathbb{R}$. Let $\sigma(X)$ be the sigma-field generated by $X$ and $\mathcal{B}( \mathbb{R})$ the Borel ...
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2answers
56 views
2
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3answers
24 views

Exponential Random Variables and either cases of a Conditional Expectation

We are given a random variable X which has an exponential distribution of parameter λ=1. $$X\sim\exp(λ=1)$$ We know that $$E[X]=\frac{1}{λ}$$ Hence for us $E[X]=1$. By virtue of the memoryless ...
2
votes
2answers
36 views

Expectation maximum between a constant and a random variable

Let $X$ be a random variable. For sake of simplicity assume it is uniformly distributed from $[0,1]$. Let $c$ be a constant in the same interval. How do I express $E[\max(X,c)]$ in such a case?
1
vote
1answer
56 views

When the two conditional expectations are independent?

Consider $X,Y$ be two independent random variable I want to know under what sigma-algebra $\mathcal{F}$, we can say the conditional expectation $E[X|\mathcal{F}]$ is independent of ...
3
votes
2answers
43 views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?