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

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1answer
22 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
48 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
33 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
15 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
33 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
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0answers
20 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
30 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
votes
1answer
51 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
23 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
25 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
19 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
36 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
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1answer
28 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
39 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
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1answer
40 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: ...
0
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1answer
40 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
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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
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0answers
14 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 ...
0
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1answer
34 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 ...
1
vote
0answers
29 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 ...
0
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0answers
20 views

Measurability and random variables [duplicate]

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 ...
0
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2answers
65 views
2
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3answers
26 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
44 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
59 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
65 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?
1
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0answers
17 views

Product of random binary vector with random binary matrix in GF(2)

Suppose we have a binary vector $f$ with dimensions $1×l$ such that each entry in the vector is generated independently with propability $q$ of being $1$. And we have a binary matrix $G$ with ...
0
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0answers
14 views

Transform two correlated random variable to independent variables without knowing correlation

I am thinking about this interesting question which arises in the following realistic setting. For example, in one medical experiment one drug and one placebo are applied to two randomized groups of ...
1
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1answer
40 views

Box-Muller Transformation

I know that we can use the Box-Muller transformation to generate a pair of independent standard Gaussian random variables using a pair of independent standard uniform random variables. I am wondering ...
1
vote
1answer
54 views

Poisson distribution and idd random variables - Proof of an equality

I want to solve the following task The first part was very easy for me. But I dont know how to solve the second one. I guess I even understood it completely. I am thankful for any kind of help!
0
votes
0answers
39 views

Indepenent variables and functions

Random variables $x_1, x_2,...,x_n$ are independent. Then, how to prove whether these functions $$y_1=f_1(x) \\ y_2=f_2(x) \\ ... \\ y_n=f_n(x)$$ are independent or not . where, $x=(x_1,...x_n)$ ...
0
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1answer
20 views

Is there any relationship between the skewness parameter in the “stable distribution” and the shape parameter in the Skew normal distribution?

While studying Stable Law we understand that when alfa (tail index parameter) is 2, then regardless of beta (skewness parameter) the random variable is normal. This may be viewed by checking the ...
1
vote
1answer
32 views

Question about “linear programming problem” in reference to joint pmf

I'm working on a homework problem and I'm not totally sure what the question is asking... The question reads: "Consider the linear programming problem: maximize $Ax_1+Bx_2$ subject to $x_1+x_2\leq ...
2
votes
1answer
23 views

Expected value of gain

The operator of a tour has a bus with 20 seats. The operator knows for experience that it can occur that not all of the tourist make it on time, so he sells 21 tickets. The probability that a tourist ...
1
vote
1answer
34 views

Traffic with Poisson distribution

The number of cars that cross an intersection during any interval of length t minutes between 3:00 pm and 4:00 pm has a Poisson distribution with mean t. Let W be the time that has passed after 3:00 ...
1
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1answer
31 views

Mean time to failure of a system problem

The problem: A system has 2 components: A and B. These components have independent lifetimes that are exponentially distributed with parameters 2 and 3 respectively. (Recall an exponential prob. ...
0
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1answer
39 views

Likelihood of a function of different types of random variables

Is there a general way of expressing the likelihood of some known, but non-trivial function of several random varaibles. For example, suppose that we need to calculate the parameters of a process ...
2
votes
1answer
28 views

What is probability that students will be evenly divided among the 3 categories? What is the marginal probability that 2 will be in the middle half?

Problem: The campus recruiter for an international conglomerate classifies the large number of students she interviews into three categories - the lower quarter, the middle half, and the upper ...
2
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0answers
26 views

The independence of random variables

Here is my question: Consider a homogeneous ergodic Markov chain on a finite state space $X=\{1,\ldots,r\} $. Define the random variables $\tau_n \,,n\ge1$ as the consecutive times when the Markov ...
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1answer
37 views

Complete convergence and almost sure convergence of random variables

Let $X_{n}$ be a sequence of independent random variables. Prove that $X_{n}$ converges to zero, almost everywhere (a.e.) if and only if for all $\epsilon >0$, $\sum_{n=1}^{\infty } ...
0
votes
1answer
16 views

Variance for sum of two correlating variables.

There are 2 random variables, X and Y. The $E(X) = -1\; and\;E(Y)=6$ I also know that $Var(X) = 6 \; and \; Var(Y) = 9 \; and \;Cor(X,Y)=0.9$ How can i calculate $Var(X+10Y)$ ? I tried to calculate ...
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vote
2answers
21 views

Showing 1/E(W) <= E(1/W)

How do I show that $\displaystyle \frac{1}{E(W)} \leq E\left(\frac{1}{W}\right)$ for a positive random variable W? I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq ...
1
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0answers
19 views

Existence of a sequence of random variables, provided weak convergence

I'm trying to prove the following statement: Let $ X_n, X_0 $ be such R.V.'s that $ X_n $ converge to $ X_0 $ in distribution (weakly). Prove that there exist $ Y_n, Y_0 $ on the probabilistic space ...
2
votes
1answer
63 views

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, ...
0
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0answers
24 views

Find a Borel function

I have trouble understanding what is a random variable. The problem arose when I wondered: Let $X$ and $Y$ be independent and equally distributed random variables. Find a Borel function $B$ such that ...
0
votes
1answer
38 views

Mutual or pairwise independence needed? Variance of a sum.

This is a simple question: Do we need mutual independence or only pairwise independence in order to state that $$\mathrm{Var}\left[\sum_{i=1}^n X_i\right] = \sum_{i=1}^n ...