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

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3
votes
2answers
41 views

Transformation on a random variable

Can someone please help me with formatting this question? $Y$ is an exponential random variable with parameter $1$. Let $Z=-Y$, what is the pdf of $Z$? Attempt: $$\Pr(-Y< y)=\Pr(Y>-y) ,$$ ...
0
votes
1answer
13 views

probability, depedence => uncorrelated?

One quick question. Two random variables are dependent, so it must be uncorrelated? Also for the terminalogy, does the term "uncorrelated" means two random variables have a corvariance factor of 0? ...
1
vote
1answer
18 views

probability, please help on bayes question

I dont need exact answer, I just need help to judge wether my following method is correct or not. Question:A physician has 5 patients. There are treatments A and B. Physician gives treament A to 3 ...
0
votes
1answer
20 views

probability, at least one is not present

you have $5$ red balls, $10$ green balls, and $15$ yellow balls in a balls. You randomly choose $5$ without replacement. What is the probability that at least one of the $3$ colors is not present ...
0
votes
0answers
21 views

Representing a randomisation function as a formula

I have formula question that I believe will use set theory notation. I'm explaining a randomisation function I developed, but I would rather explain it explicitly with formula notation, rather than ...
0
votes
1answer
17 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
1
vote
0answers
8 views

Random Variables in Matlab

Is there a way to deal with random variables as objects in Matlab? For example, suppose I define the i.i.d. random variables $X_1,\ldots, X_n\in N(0,1)$, where $N(0,1)$ is the normal distribution ...
0
votes
0answers
5 views

deterministic limit of gaussian distribution

Let $a$ be a random variable over some set $A$, and let $\mathcal A \subseteq A$ be an event. Let $\mathcal E \subset \mathbb R^n$ be another event, and let $x_1, \dots, x_n$ be several Gaussian ...
0
votes
1answer
22 views

What is the definition of sigma field generated by random variable $X$? [on hold]

What is the definition of $\sigma$-field generated by a random variable $X$? And what does it mean?
2
votes
1answer
38 views

what is the distributions of the random variable?

If moment generating function is $m(t)=[(1/3)e^{t}+(2/3)]^{5}$, then what is the distributions of the random variable?
2
votes
3answers
68 views

Probability, random line up

Five distinct families arrive to a party. Each family consists of 3 people. The 15 participants of the party are arranged randomly in a line. Let X be the number of families that their members sit ...
1
vote
1answer
25 views

probability, need help on the marginal densities

I need help on the marginal densities. In particular, I know you just integrate the joint pdf f(x,y) from y=-infinity to +infinity, but in the context of the below question, I have trouble to define ...
4
votes
0answers
129 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
1
vote
0answers
97 views

PDF of X +Y + X* Y, when X and Y are independent Normal [on hold]

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
0
votes
0answers
5 views

Is it possible to calculate the autocovariance of a DT WSS signal knowing only it's mean and it's linear estimator?

Let's also assume that we know that $C_{xx} [0] = K$ Where $K$ is some constant. I'm trying to figure out if that's possible and if yes how I would get around doing it.
0
votes
2answers
25 views

Probability Density Function with continuous random variables

Let $X$ have density $$ f_X(x) = \begin{cases} \sqrt{3(x+2)}/6 & -2 \leq x \leq 1 \\ 0 & \text{otherwise}. \end{cases} $$ Find the probability that $X$ is positive. Would this just ...
1
vote
1answer
57 views

probabilty, random variable independent

Let $X$ and $Y$ be independent Poisson random random variables with ($\lambda=1$). Are $X-Y$ and $X+Y$ independent? Justify My attempt: $X-Y$ => random variable is $0$. $X+Y$=> Poisson of ...
0
votes
2answers
23 views

probability, indicator random variable

Let $A,B,C$ be independent events with $P(A)=P(B)=P(C)=\dfrac{1}{2}$. Let $X$ be the indicator r.v. of the event $A \cup B$ and $Y$ the indicator r.v. of the event $B \cup C$. Compute ${\bf E}[XY]$. ...
-1
votes
0answers
13 views

Quotient of random variable's moments.

Let $X\sim\Gamma(k,1)$ and $Y\sim\Gamma (n+1-k,1)$ be independent random variables, where $k,n \in \mathbb{N}$ and $1\leq k\leq n$. What are the values of $$\mathbb{E}\left[\frac{X}{X+Y}\right]\text{ ...
2
votes
0answers
43 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
0
votes
0answers
10 views

Expectation of modulus of normal distribution.

I consider random variable $\xi \in N(o, \sigma^2) $. How to find teh expectation: $\mathbb{E}(|\xi|)$? It seems to be connected with the variance of $\xi$, but in which way?
0
votes
1answer
18 views

Khinchin's weak law of large numbers: finite variance

I have the following situation: suppose you have a sequence of i.i.d. random variables $\{X_i\}$ with mean $\mu$ and variance $1$. I would like to use Khinchin's WLLN on it, but this requires that ...
0
votes
3answers
286 views

Entropy of geometric random variable?

I am wondering how to derive the entropy of a geometric random variable? Or where I can find some proof/derivation? I tried to search online, but seems not much resources is available. Here is the ...
0
votes
1answer
20 views

Probability: arithmetic on Random Variables

I have a question about the arithmetic on random variable in probability. Question: Are the events $\{X=Y\}$, $\{Y=Z\}$,$\{Z=X\}$ independent? My solution: $$ P(X=Y,Y=Z,Z=X) = {(0.5^2 ...
0
votes
0answers
16 views

bound of | E[X/Y] - E[X]/E[Y] |

Is there some bound for $ | \mathbb{E}[X/Y] - \mathbb{E}[X]/\mathbb{E}[Y] | $ ? where $X$ and $Y$ are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is ...
1
vote
0answers
52 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
0
votes
0answers
9 views

How to generate a random variable with this trigonometric PDF

I need to generate random numbers $X$ with distribution $$f_X(x)\propto \sec^4(x)$$ for $x\in(0,x_m)$ where $x_m<\frac\pi2$. Any ideas on how to do this efficiently? I know the basic theory on how ...
-5
votes
0answers
32 views

Help!!!!!!!!!!!!!! [closed]

1. Assume that books from a certain publisher have an average of one misprint every 20 pages. (a) What is the probability that a given page has two or more misprints? (b) What is the probability that ...
0
votes
1answer
7 views

iid random variables (vectors)

If $(X_{1},Y_{1}), (X_{2}, Y_{2}),...,(X_{n}, Y_{n})$ denote a sequence of iid random variables from $(X,Y)$, can I say that each $X_{i}$ is independent from each $Y_{i}$? Or is it just for the ...
0
votes
2answers
31 views

probability: arithmetic with random variables.

I have a question of using arithmetic on random variables. Please refer to the following question, to which I will present my solution using the arithmetic (which I thought it's correct but actually ...
0
votes
1answer
14 views

Distribution of a function of a uniform random variable.

I ran across this example the other day and was surprised at how stumped I was. Suppose $U$ is a uniform random variable on the interval $[0,1]$. Let $F = \frac{1}{U+3}$. What is: ...
0
votes
1answer
24 views

calculating X, Y, Z random variables

Suppose X, Y, and Z are random variables that each take the value 0 or 1. If P(X=0,Y=1,Z=0)=1/3 and P(X=0,Y=1,Z=1)=1/4, what is the value of P(X=0,Y=1)? I am trying to calculate this but I am really ...
0
votes
1answer
21 views

calculating variance of a random variable

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other three ...
-1
votes
0answers
39 views

A basic question on characteristic function

Suppose I have two random variables $X$ and $Y$ for which characteristic functions are same. Let $F$ and $G$ be their distribution functions. I have to prove that $F$ and $G$ have the same set of ...
2
votes
1answer
39 views

Finding Random variables measurable

If [0,1] is our sample space and our sigma algebra is generated by all segments of the form [0,2^(-n)]. How can we describe the random variables measurable with respect to our sigma algebra? I'm ...
0
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0answers
13 views

Non-Linear Transformation of two R.V.

Suppose I want to take some action if an object is with in some distance from me with some probability. I have an x and y measurement of its position with noise ...
1
vote
1answer
32 views

Calculating bounds with multiple random variables.

I have this problem: Suppose there are 4 students (who we'll refer to as A, B, C, and D) in a class and each student is equally likely to have been born in any of the twelve months of the year. For ...
0
votes
2answers
25 views

Covariance of dependent random variables from a Poisson process

Question: Given a Poisson process $N(t),t≥0$ with rate $λ$, calculate the covariance of $N(2)$ and $N(3)$. Attempt: So clearly $N(2) \sim Po(2\lambda)$ and $N(3) \sim Po(3\lambda)$. So, ...
0
votes
0answers
19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
1
vote
0answers
57 views

Copulas/Probability Theory

So I have a basic understanding of copulas but wanted to verify I'm applying things correctly to reach the correct outcomes.. Show that as $\theta\to\infty, C^{Fr}(u_1,u_2)\to\min(u_1,u_2)$, the ...
0
votes
1answer
47 views

Fair Die tossed twice, random variables

A fair die is tossed twice. Let $d_1=\text{value of die on roll 1}$ and $d_2=\text{value of die on roll 2}$ Let $X=d_1+d_2$, the sum of the faces; $Y=\max\left\{d_1,d_2\right\}$, the maximum of the ...
-1
votes
0answers
33 views

If X is a Poisson variate and $ p(X = 3) > p(X = 2)$ [closed]

$X$ is a Poisson variate and $ p(X = 3) > p(X = 2)$ Then how to find the the minimum value of the mean.
1
vote
1answer
34 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
0
votes
1answer
36 views

probability of playing music player on shuffle and listening to every song.

I have a few problems I am trying to work out but I am not totally confident in my answers: The problem is such: Suppose you have a playlist consisting of four songs. You play your playlist in ...
1
vote
0answers
27 views

Stochastic domination by coupling

The following is a slightly streamlined version of Exercise 7.5 in Dubashi & Panconesi's "Concentration of Measure for the Analysis of Random Algorithms": Let $X$ and $Z$ be independent random ...
0
votes
1answer
231 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
0
votes
1answer
26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
2
votes
2answers
47 views

Does a proportion have to be a rational number?

Does a proportion have to be a rational number? For example, Assume we have a square with side $2$ units. We are throwing a circle of radius $1$ unit over the square. Let $X$ be the area of the ...
1
vote
1answer
26 views

Finding density function of random variable, which is division of two other random variables.

I have following 2-dimensional random variable $(x,y)$: $$ f(x,y) = 1, \quad 0 \leq x \leq 1, \quad 0 < y \leq 1 $$ I have to find density function of random variable $Z = \frac{X}{Y}$. I am ...
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votes
0answers
31 views

Matrix of Expectation of Random variables Update [closed]

I am not a math guy, but here I have encounter a problem about finding an inverse matrix, which the original matrix are elements of expectation of random variables. I think it is an optimization ...