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

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0answers
12 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 ...
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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 ...
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1answer
20 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?
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1answer
33 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?
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1answer
23 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 ...
2
votes
3answers
66 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 ...
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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.
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0answers
82 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!
1
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1answer
52 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 ...
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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 ...
0
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2answers
22 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 ${\bf E}[XY]$. ...
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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?
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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 ...
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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{ ...
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1answer
16 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 ...
2
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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 ...
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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 ...
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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 ...
1
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0answers
49 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 ...
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0answers
32 views

Help!!!!!!!!!!!!!! [on hold]

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 ...
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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 ...
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2answers
30 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 ...
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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: ...
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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 ...
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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 ...
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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
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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 ...
2
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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 ...
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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
2answers
24 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, ...
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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 ...
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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 ...
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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
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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
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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 ...
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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 ...
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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
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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 ...
0
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1answer
51 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
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1answer
11 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
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0answers
30 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 ...
0
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1answer
27 views

probability: chebychev inequality question

For this question, I don't understand the highlighted part of the solution I thought it should be >5, but then 6?
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0answers
21 views

Error in thinking: Poisson Process is a Markov Process

I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$: $P(X_{t_n} = a_n | ...
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2answers
34 views

Prove that a constant multiplied by a Poisson random variable is not Poisson

Does the following constitute a proof that the multiplication of a Poisson random variable $K$ with an integer constant $a$ is not itself Poisson? That is, $f_K(k) = \frac{\lambda^k}{k!} ...
2
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1answer
32 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
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3answers
35 views

Independence of max and min of a set of random variables.

Suppose $X_1,\ldots,X_n$ are independent and identically distributed random variables with cdf $F_X(x)$. Define $U$ and $L$ as $U=\max\{ X_1, \ldots ,X_n\}$ and $L = \min\{X_1,\ldots,X_n\}$. Are $U$ ...
1
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2answers
72 views

Random Variable Probabilities

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 four ...
0
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0answers
10 views

linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
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2answers
47 views

Is this probability inequality always true?

For $n$ random variables $X_1$, $X_2$, $\dotsc$, and $X_n$. Is it always true that: $$\mathbb{P}\left[\sum_{k=1}^{n} X_k>a\right]\geq\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}>a\right].$$ ...