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

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1
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1answer
21 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 ...
1
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1answer
21 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 ...
0
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1answer
33 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 ...
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1answer
27 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. ...
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1answer
29 views

Continuous Random Variables Homework [closed]

Suppose $f(x)$ is the density function of a continuous random variable $X$ with support on the interval $(-1,1)$. a. Express $P(X \leq x)$ in terms of the function $f$. b. Write an integral ...
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0answers
23 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
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1answer
23 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
24 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 ...
1
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1answer
29 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
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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 ...
1
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2answers
19 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 ...
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0answers
17 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 ...
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1answer
26 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
18 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
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1answer
32 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
21 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
185 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
23 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
84 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
69 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 ...
1
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2answers
43 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
30 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
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2answers
30 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
26 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 ...
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1answer
11 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: $$ ...
0
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2answers
26 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
29 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
35 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 ...
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0answers
21 views

Estimation in Multiplicative and additive Noise problem

I have unsolved problems below for random process homework. Consider the problem of estimating a random vector $\underline{S}(u)$ from an observation $\underline{X}(u)$, where ...
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0answers
5 views

Lag Autocovariance

Can somebody tell me what is the importance of calculating the lag autocovariance matrix, and what information I can extract from it?
0
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1answer
17 views

Density function for RV X

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
48 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 ...
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0answers
50 views

Expectation of geometric random variable

Let $X$ be geometric random variable with parameter $p$. How to prove that: (1) $E[X-1|X>1] = E[X]$ (2) $E[X^2|X>1] = E[(X+1)^2]$ Author explains the fact below and states it is used ...
8
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3answers
148 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
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0answers
13 views

Density function for RV

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 ...
1
vote
1answer
27 views

Given pdf $f_{Y}(y)$ = $ay^2e^{-by^2}$, $y >=0$, find the pdf of the kinetic energy, $W = 0.5mY^2$

problem: Suppose the velocity of a gas molecule of mass $m$ is a random variable with pdf $f_{Y}(y)$ = $ay^2e^{-by^2}$, $y >=0$, where $a$ and $b$ are positive constants depending on the gas. Find ...
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0answers
40 views

Can someone explain this question about random variables?

In each of the following questions, a random variable is described. Identify the random variable as either binomial, Poisson, geometric, negative binomial or hypergeometric and describe a reasonable ...
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1answer
32 views

Coupling of r.v.

I am trying to answer this question. If $X$ and $Y$ are random variables on $(\Omega, \mathcal{B})$, show \begin{align*} \sup_{A \in \mathcal{B}} |P[X \in A] -P[Y \in A]| \le P[X \neq Y]. ...
2
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0answers
30 views

Sum of $\{X_n\}$ iid random variables contained in a compact interval implies each $X_i=0$ a.s.?

I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$, $$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$ Does this require $X_i ...
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0answers
16 views

Constraint Optimization

I have a sequence of iids defined by: $f(x|\theta) = \exp(-(x-\theta))\;\;\;\; \theta<x<\infty$ To find the maximum likelihood estimate, i should maximize the log likelihood with respect to ...
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1answer
11 views

Probability in continuous functions. ( Simple Question )

Completed the question but can't get my head around part e? Can someone help?
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0answers
14 views

Conditional independent random variable in graphical models vs measure theory

When we say random random variables $X$ and $Y$ are conditionally independent given and $Z$, I understand it to mean given the $\sigma$ algebras generated by $X$,$Y$ and $Z$ denoted ...
0
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1answer
32 views

How can I prove that Xn converges to 0 in distribution?

Xn~U[-1/n,1/n]. Since for convergence in distribution Xn-->X iff Fn(x)-->F(x). First of all, I am trying to get the cumulative and it is Fn(x)=(1+xn)/2. Hence, am I doing something wrong?
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0answers
19 views

What is the closed-form expression for the energy of this term?

Is there a closed-form expression for the energy of the following term? $r_k = \sum_l \sum_i a(i) h(l-i) c(k-l)$ that is $E(\sum_l \sum_i a(i) h(l-i) c(k-l))^2$ where $ a \sim N(0,\sigma ^2)$ and $h$ ...
1
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2answers
85 views

Independent Exponential Random Variables

I am currently trying to figure out a problem and it is using notation that I have never seen before so I am pretty stuck, any suggestions would be greatly appreciated! Let $X, Y, Z$ be independent ...
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0answers
19 views

$P(X\le z)$ for $z$ between $z\in(1,\infty)$ for binomial distribution

If I have a binomial distribution of $X$ (a random variable), where $X=\{I:X_1= \dots X_{i-1}=0, X_i = 1\}$, how do I find an expression $P(X\le z)$ for $z\in(1,\infty)$? Any help appreciated!
0
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1answer
16 views

Upper bound of min separation between n randomly chosen points in a figure

Given a rectangle of size $6\times12$, prove that if $7$ points in it are chosen uniformly at random, the distance between at least $2$ points is $\leq 5$. I don't know how to approach this problem. ...
0
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1answer
22 views

Random sample of size $n = 2k$, calculate $p(X_1 < 1/2, X_2 > 1/2, X_3 < 1/2, X_4 > 1/2, \dots, X_{2k} > 1/2)$.

A random sample of size n = 2k is taken from a uniform pdf defined over the unit interval. Calculate $p(X_1 < 1/2, X_2 > 1/2, X_3 < 1/2, X_4 > 1/2,...., X_{2k} > 1/2)$. Solution: Then ...
2
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1answer
57 views

Good ways to sample $n$ identical and dependent random variables

I'm wondering if there's a good way to talk about sampling identical but dependent random variables where it's also easy to see how the distribution evolves as we move from $n$ random variables to ...