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

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0
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0answers
7 views

how to prove independence in this case

The question is : $X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent. What I ...
0
votes
1answer
18 views

Notation in sequences of random variables

I read in statistics books that a sequence of random variables are often written as ${X_n}$. But in all the theorems it just says $X_n$. Why is that? And does ${X_n}$ symbolize the WHOLE sequence ...
0
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0answers
4 views

A derivation of expected values of difference of random processes

Hope to ask a calculation step from a paper: Let $\mathcal{S}$ be a subset in the Euclidean space $\mathbb{R}^n$. Let $x \in \mathcal{S}$. Let $y \in\mathcal{S}_2^{k-1} = \{y \in \mathbb{R}^k ...
0
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0answers
21 views

How to understand a complicated random process

I read a paper and it defines a r.p as following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^k$ $\{h_i\}_{i=1}^n$ and $\{g_j\}_{j=1}^k$ are two indep. sets of orthonormal r.v.'s Define a r.p: ...
2
votes
2answers
53 views

Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
1
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0answers
15 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
0
votes
1answer
16 views

Central Limit Theorem & Delta method problem

Let $U_1$,...,$U_n$ be a random sample from the U(0,1) a. Let $X$=-log($U$). Find the distribution of X b. Let $Y$=$1/{\prod_{i=1}^n U_i^{1/n}}$, where $U_1$,...,$U_n$ be a random sample from the ...
0
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0answers
37 views

Expectation and variance of a comboflip

Here's the question, a comboflip is the simultaneous flip of a fair coin and toss of a fair die. Comboflips are done until at least one head has been seen and at least one 6 has been seen(they do not ...
0
votes
2answers
33 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
0
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0answers
28 views

Continuous random variable - probability density function question

A continuous random variable $X$ has a single parameter $a$. The probability density function of $X$ is $f(x)=c(1-x^2)$ for $-a<x<a$ and some constant $c>0$. a) What are the allowable values ...
1
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1answer
11 views

Notation for image of a discrete random variable?

Suppose we have a discrete probability space $(\Omega,\Sigma,\mathbb{R})$ and a discrete random variable $X:\Omega \to \mathbb{R}$. A usual way to denote the set of values that $X$ takes is simply ...
0
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1answer
10 views

Proving something is strict stationary

Let $W$ be a uniform distribution on $(0,\pi)$. Let $Z_t=\cos(tw)$. I know that $Z_t$ is a strict stationary but I have no idea how to prove this. Can someone give me some methods?
1
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2answers
28 views

Expectation of a function of pairs of random variables

For positive random variables $(X_1, Y_1)$ and $(X_2, Y_2)$, suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ have the same distribution and (the two pairs) are independent. Also suppose that $E[Y_1|X_1] = ...
0
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1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
-1
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1answer
18 views

uniform convergence of random variables [on hold]

Consider a continuous function $f:[a, b] \rightarrow \mathbb{R}$. Let $U_{1}, U_{2}, \cdots$ independent and identically distributed with $U_{k}\sim U([a,b])$. Show that: $$ \dfrac{f(U_{1}) +f(U_{2}) ...
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0answers
25 views

function of a random variable problem 2 [on hold]

Let $U$ be a continuous random variable with uniform distribution over $[0,1]$. Define $X$ by $$X=\operatorname{Int}\left(\frac{\ln(1-U)}{\ln(1-p)}\right)+1$$ where $\operatorname{Int}(x)$ is the ...
0
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1answer
25 views

Uniformity of the difference between two random variables

What can I say about the distribution of two random variables $A$ and $B$ such that $A-B$ is uniformly distributed?
-3
votes
2answers
65 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
0
votes
1answer
30 views

Explicit CDF associated to Gamma PDF [on hold]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...
2
votes
1answer
81 views

Evaluate this covariance matrix.

Edit: I have added an approach provided by @GiannisChantas. It would be great (and much appreciated) if someone could check if this approach is correct! I have also added a secondary question for ...
1
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2answers
42 views

Find $E[N]$, where $N = \min\{n>0: X_n = X_0\}$

Let $X_i$, $i\geq 0$ be independent and identically distributed random variables with probability mass function $$ p(j) = P\{X_i=j\},\; j=1,...,m,\;\sum^{m}_{j=1}P(j)=1 $$ Find $E[N]$, where ...
0
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0answers
11 views

discrete random variable

The whole question is: Let X be a discrete random variable and let Y = 0.5 X + 3. (i) Assume that the PMF of X is given by enter image description here where k is some suitable constant. Determine ...
-2
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0answers
21 views

Deriving the Pareto Distribution from an Exponential Distribution [closed]

Let $T$ be an exponential random variable with hazard rate $a>1$. Consider a random variable defined by the condition $X = b (e^t - 1)$." I need to find the density of $x$.The answer is ...
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votes
1answer
22 views

Get unknown value in discrete random variable

Let $X$ be a discrete random variable (i) Assume that the PMF of $X$ is given by $$\operatorname{Pr}(X=x)=\begin{cases}kx^{2} & x \in \{-4,-2,0,2,4\} \\ 0 & x\not\in \{-4, -2, 0, 2, ...
0
votes
0answers
11 views

Central Limit Theorem for independent but non identically distributed random variables

My question is the following: Given the sum of R.V.s, $Z_N = X_1 + X_2 + ... +X_N $, where $X_i$ are independent, Rice distributed ($X_i\sim Rice(\mu_i,\sigma) $), is there any way to approximate ...
0
votes
2answers
44 views

Probabilities and random variable problem

Suppose we have a random variable X, and we are given the numerical values of its expectation as well as its s.d. (standard deviation). How can I go about finding the maximum value the probability of ...
0
votes
1answer
28 views

What is the meaning of E and d in this formula?

I am trying to learn the information bottleneck method. On slide 15, they give this equation. I think I understand that X is a random variable (but do not understand the meaning of the exponent, n). I ...
0
votes
3answers
33 views

Generate random variable from series of its expected values E[X], E[X^2], E[X^3], …?

Given a series of all the expected values of a random variable, can we find the random variable itself ?
1
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1answer
22 views

Find the cdf associated with each pdf (NOT transformation)

Find the cdc associated with each pdf: a) f(x) = 3(1-x)^2 , 0 < x < 1 , zero elsewhere b) f(x) = 1/x^2 , -infinity < x < infinity The answers are a) 1-(1-x)^3 , 0 <= x < 1 b) 1 ...
1
vote
0answers
22 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
0
votes
1answer
37 views

Variational problem concerning variances

Let $\phi$ be the family consisting of all random variables $X$ such that $P(X\in [0,1])=1$, $EX=\frac{1}{3}$, $P(X<\frac{1}{4})<\frac{1}{2}$, $P(X>\frac{1}{4})\geq\frac{1}{2}$. Calculate ...
1
vote
1answer
21 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
0
votes
1answer
12 views

Two random variables X,Y are, X,Y independant b. are X+Y X-Y independant

if X and Y are independent, check whether the (0, 0) value is the same as P(X=0) P(Y= 0), and the same with the other 4 entries. Make a table with the distributions of X + Y and X - Y. For any ...
1
vote
1answer
49 views

A fair die is tossed until the sequence “44” is seen. Let N be the number of tosses this requires. Find E $N$

A fair die is tossed until the sequence “44” is seen. Let $N$ be the number of tosses this requires. Find $E[N]$. I have my own solution which I need someone to verify. And this problem has to be ...
1
vote
2answers
39 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
-1
votes
1answer
29 views

Inequality of expectation of random variables [closed]

Let $X$ and $Y$ be two continuous random variables in $\mathbb{R}$. $X$ and $Y$ may not have any kind of stochastic dominance properties. Let $f$ be a strictly monotonically increasing concave ...
1
vote
2answers
97 views

Can the expected value of a PMF be zero, as in E[X] = 0?

The whole question is: Let X be a discrete random variable and let Y = 0.5 X + 3. (i) Assume that the PMF of X is given by where k is some suitable constant. Determine the value of k. (ii) Find E ...
0
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0answers
7 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
0
votes
1answer
39 views

If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable ...
2
votes
1answer
28 views

Joint density function with absolute value

Let X and Y have joint density fXY (x, y) = kxy^2 where j0 ≤ x, y ≤ 1, 0 otherwise. Compute Pr(|X − Y | < 0.5). So I found that k=6, but can't figure out the probability part after working on ...
2
votes
1answer
28 views

Is this a Markov chain? [duplicate]

Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$. $S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$ $Y_n = \sum\limits_{i=0}^{n} S_i$. My ...
1
vote
2answers
18 views

PMF of X: Number of trials to draw a chip

Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red ...
0
votes
2answers
71 views

PMF of number of heads of 4 coin tosses

Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number. I initially ...
0
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0answers
20 views

Let $X$ be uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$.

Let $X$ be a continuous random variable uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$. $P(X-X^2 \leq a)= P ( -X^2 + X - a \leq 0) = P ( -X^2 + X - a \leq 0| ...
0
votes
2answers
26 views

If $P(X \geq k) = p^k$, for $k=0, 1, 2,…$ then $P(X=k)=p^k(1-p)$

If $P(X \geq k) = p^k$, for $k=0, 1, 2,...$ then $P(X=k)=p^k(1-p)$ The converse is immediate but I don't know how to approach the direct implication.
0
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1answer
18 views

Transformation of Random Variable results in strange CDF

I'm trying to transform a RV according to $Y=X^{-a}$ with $a>0$ and X being uniformly distributed in $[0,A]$: $ F_X(x) = \begin{cases}0 & x<0 \\ x/A & 0\leq x \leq A \\ 1 ...
2
votes
1answer
61 views

If $X$ is Poisson, find the expectation of $\frac{1}{a+X}$

If $X$ is a Poisson random variable with $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$ and $a>0$ then find the expectation of $\frac{1}{a+X}$ If I make use of ...
1
vote
1answer
25 views

How does $\mathcal{L}^1$-convergence of a series of $\mathcal{L}^1$ random variables imply that $\sup_{n \in \mathbb{N}} \mathbb{E}[|X_n|] < \infty$?

Let $(X_n)_{n \in \mathbb{N}}$ be a series of random variables with $\forall i: X_i \in \mathcal{L}^1(\Omega, \mathfrak{F}, P)$ and $X_n \rightarrow^{\mathcal{L}^1}X$. How do I show then, that ...
0
votes
1answer
44 views

Random variable modeling arthroscopic meniscal repair

The below problem is from my introductory stats textbook, the chapter on random variables and probability distributions. I don't even know what's being asked, much less how to answer it. Any clues? ...
-1
votes
1answer
33 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...