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

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1
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0answers
9 views

Bound the variance of the product of two random varables.

For two random variables $X$ and $Y$ show that the following inequality holds $$Var(XY)\leq 2\|Y\|_{\infty}^{2}Var(X)+2\|X\|_{\infty}^{2}Var(Y).$$ Well first I tried to show it for just indicators ...
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2answers
22 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
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1answer
8 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 $Y$ ~ $Bin(m,p)$, ...
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1answer
20 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 ...
2
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0answers
9 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
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0answers
11 views

Expected of greatest sum in an uniform distribution taking only 2 values

What is the expected value of the probability to take two numbers that the sum of their values in an uniform distribution in [0,1] of n random variables is the greatest (x1,x2,x3,x4,...,xn)?
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0answers
7 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 ...
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0answers
16 views

Lognormal Random Variable [on hold]

Let $S(n)$ denote the price of stock $n$ days from now. An investor believes that the price ration is $S(n)/S(n-1)$ are independent and identically distributed lognormal random variable with ...
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0answers
8 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 ...
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1answer
12 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
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1answer
18 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
54 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
15 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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votes
2answers
20 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
18 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
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1answer
23 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
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0answers
26 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 ...
1
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0answers
13 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
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2answers
62 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
64 views

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

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
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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 ...
0
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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
24 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
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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)) ...
0
votes
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 ...
0
votes
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 ...
0
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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
votes
1answer
29 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 ...
0
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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 ...
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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
votes
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: $$ \mathbb{E}[X_1|X_1 + ... + X_n = x] $$
1
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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 ...
0
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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
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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
14 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
votes
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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votes
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...