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

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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
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 ...
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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
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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
31 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
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0answers
27 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
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 ...
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2answers
65 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
39 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?
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1answer
58 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
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2answers
48 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?
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0answers
15 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
39 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
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1answer
52 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!
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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, $x=(x_1,...x_n)$ ...
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0answers
3 views

LR- Fuzzy Random Variable - Intuitive explanation

What is difference between Kwakernaak and Puri's view of FRV? From which category LR-Fuzzy Random Variable belongs?
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1answer
16 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 ...
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0answers
10 views

random variables stochastically bound problem

Could you help me about stochastically bound problem for random variable. show that, there exist a sequence of {a_n} of positive real numbers such that X/a_n->0 a.s for any random variable X.
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1answer
28 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
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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
30 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
36 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
26 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
25 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
35 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
15 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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2answers
20 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
18 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 ...
1
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1answer
34 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, ...
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0answers
20 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
34 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 ...
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0answers
24 views

Order statistics of mixed (iid as well as non iid) random variables

Does anyone know if there are results (PDF or the CDF) on the order statistics (at least minimum or maximum) of $n$ random variables in which a few of them are i.i.d. and the rest of them are ...
6
votes
2answers
217 views

$\{X_n\}$ are iid random variables with symmetric distribution

Let $X_1,X_2,\ldots,X_n$ be iid random variables with symmetric distribution. Show that $$P\left(|X_1+X_2+\cdots+X_n|\ge \max_{1\le i\le n}|X_i|\right)\ge \frac12.$$ I was trying it for $n=2$. ...
1
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1answer
25 views

two soft questions about stochastical ordering

I have two questions and I will be very happy to hear your comments: a-) For two random variables $X$ and $Y$ let $X$ dominate $Y$, i.e. $X>_{ST}Y$. let $f$ be a positive function. Is it true ...
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0answers
87 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
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3answers
79 views

Probability of two normal random variables when random samples are taken from a population

This is sort of second section to my previous question, I should have included both together, but I forgot to. Sorry for any inconvenience. X= random height of a male Y= random height of a female X ...
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2answers
45 views

If $g$ is a function of the random variable $X$, is it true that $H(X) = H(X) + H(g(X)\mid X)$?

I think my homework about entropy is formulated incorrectly. The question is the following: let $X$ be a discrete random variable. Show that the entropy of a function $g$ of $X$ is less than or ...
0
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1answer
21 views

How can I find the density function of Z?

I am trying to find the density function for Z, this is what I am doing but I am not getting an appropiate function, I don´t know if there is something wrong with limits of the intregral. Or if this ...
2
votes
1answer
31 views

Conditional expectations and random vectors.

Let $(\Omega, \sigma,P)$ a probability space and $Y$ a random variable on it, with $E|Y|<\infty$. Let $X_1,X_2$ random vectors with $\sigma(Y,X_1)$ independent of $\sigma(X_2)$. The problem is to ...
2
votes
2answers
33 views

Find the probability that a geometric random variable $X$ is an even number

Let $\alpha$ be the probability that a geometric random variable $X$ with parameter p is an even number a) Find $\alpha$ using the identity $\alpha=\sum_{i=1}^{\infty}P[X=2i]$ b)Find $\alpha$ by ...
0
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1answer
29 views

Random Process, how do I understand this?

I think I have little difficulty in understanding the "Random Process". Here is a definition taken from Oppenheim's book. In Section 7.3 we defined a random variable X as a function that maps ...
0
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1answer
13 views

$P_{X1,X2}(x_1,x_2)={x_1x_2 \over 36}$ find joint and marginal of $Y_1$

Let$$P_{X_1,X_2}(x_1,x_2)={x_1x_2 \over 36}, \text{ }x_1,x_2=1,2,3$$ $$Y_1=X_1X_2,Y_2=X_2$$ Find: 1) The joint PMF of $Y_1$ and $Y_2$ 2) The marginal PMF of $Y_1$ What I got: $$ ...
1
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2answers
28 views

Easy Probability Distribution Problem

Suppose we randomly draw 20 winning numbers out of 70 numbers. Let $X_m$ be the number of winning numbers that we got when choosing $m$ numbers. Determine the distribution of $X_m$, i.e. $P(X_m = k)$ ...
0
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2answers
32 views

Continuous Bivariate Random Variable, Conditional Probability Problem

I am trying to study Bivariate Random Variables. The question is if joint pdf is given by $$ f(x,y) = \begin{cases} 8xy & 0<x<1 \hspace{2mm}\text{ and }\hspace{2mm} 0<y<x \\ 0 ...
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0answers
37 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
0
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1answer
18 views

Density function for RV X [duplicate]

The density function for a random variable X is given in terms of a constant c. Find the value of c. What is the corresponding distribution function? Sketch both the density and the distribution ...
2
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1answer
54 views

Prove that $S_n^2-s_n^2$ is martingale

Let $(X_i)$ be iid such that $EX_i = 0$ and $\operatorname{Var}X_i = \sigma_i^2$. Let $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $S_n = \sum_{i=1}^n X_i$. Prove that $S_n^2 - s_n^2 $ is martingale. My ...