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

learn more… | top users | synonyms

0
votes
0answers
27 views

Approximately identical distributions

Suppose we have two random variables $x,y$ in $\mathbb{R}^n$. Assume that for some scalar $\epsilon>0$, for any set $S\subset\mathbb{R}^n$ there exists sets $S_1\supset S,S_2\supset S$ such that ...
0
votes
1answer
33 views

What do I plug in for the z score?

Cough-a-Lot children’s cough syrup is supposed to contain 6 ounces of medicine per bottle. However since the filling machine is not always precise, there can be variation from bottle to bottle. The ...
0
votes
1answer
24 views

Proving independence of functions of independent variables.

How would you go about proving that two random variables are independent, when the variables are not both discrete or continuous? Also how would you then prove functions of these random variables and ...
1
vote
1answer
19 views

When is the bootstrap sampling method not applicable?

I have used once the bootstrap sampling method to obtain a confidence interval for the expected daily returns that I had calculated using some data given. As far as I have understood, this method can ...
1
vote
1answer
16 views

Probability that a continuos r.v. lies in a certain interval, but in which function: pdf or cdf?

I was reading about random variables, both discrete and continuos, and I was reflecting about what it means for a continuos random variable to lie in a certain interval. My doubt or suspect was about ...
0
votes
2answers
50 views

how to calculate the P(the sum of upward face after 7 dice rolls <= 30)

I'm new to the stackexchange math community so forgive me if there are problems with the question format or the question itself. I'm new to probability and is trying to learn the concept of ...
2
votes
1answer
47 views

Double integration proof for expectation of a random variable.

Let X $\ge 0$ be a continious random variable. Prove $E(X) = \int_0^\infty P (X>t)dt$. Here is the proof: Denote by f the density of X, then $\int_0^\infty P(X > t)dt = ...
1
vote
1answer
19 views

Can there be metrics on sets of random variables?

First off - I do not know much probability theory, so please pardon me if this question is nonsensical. The question arose from the following thought: can I make the expectation function continuous, ...
1
vote
1answer
45 views

If $A $ and $B$ are independent R.V., will $A^2$ and $B$ be independent? [duplicate]

Given A,B are independent identically distributed random variables will $$E[A^2B]=E[A^2]E[B]$$ (uncorrelation) $$P_{A^2B}(a,b)=P_{A^2}(a)P_B(b)$$ (independence) hold? An initial thought is that ...
1
vote
1answer
24 views

Correlation and Covariance

The book I'm reading gives this as an example for lognormal variables. Starting at some fixed time, let $S(n)$ denote the price of a security at the end of $n$ additional weeks, $n \ge 1$. A popular ...
1
vote
1answer
36 views

About convergence in probability

A few days ago, I was introduced the concept of convergence in probability, after the almost sure convergence. I understood the almost sure convergence (I think): We have a sequence of random ...
3
votes
2answers
49 views

Let $|X| ≤ b $ be a random variable. Show $\forall a \in [0,b)$ $ \mathbb{P}(|X|>a)≥\frac{\mathbb{E}[X]-a}{b-a}$

I've been set the following question, but I think the question is wrong. Let $|X| ≤ b $ be a random variable. Show $\forall a \in [0,b)$ $ \mathbb{P}(|X|>a)≥\frac{\mathbb{E}[X]-a}{b-a}$ ...
2
votes
3answers
38 views

Convergence of independent $\mathcal U {(n,n^2)}$ random variables?

What does this sequence of random variable's distribution converge to? The random variables are given as follows $$Y_n=\frac{X_n-n}{n^2}, \quad n=1,2,3, \dots$$ and $X_n\sim\mathcal{U(n,n^2)}$- a ...
2
votes
1answer
35 views

Sequence of Martingales convergent in $L^1$-norm

Suppose $X^n_t$ is a sequence of martingales on a filtered probability space $\left(\Omega,\mathcal{F},\mathbb{P},\left(\mathcal{F_t}\right)_{t\in\left[0,T\right]}\right)$, that is for $\Delta>0$ ...
1
vote
0answers
30 views

Book on limit theorems for sums of random variables

I'm currently studying for an advanced course of probability theory and I would want to find a book which explains clearly limit theorems for sums of random variables. We used characteristic ...
0
votes
1answer
22 views

Need Help explaining this equation for the correlation coefficient

As I am not a math geek so I have problem comprehending this equation: equation for the correlation coefficient Its basically the formula used in the CORREL functon in Microsoft Excel and I am ...
0
votes
0answers
21 views

Rate Distortion function for vector gaussian sources

Let us consider a Gaussian source with output $\bar{X} = [X_1, X_2, ..., X_M]$ where $X_m$ are independent gaussian random variables and $X_m$ has the variance $N_m$. Suppose the per-source-letter ...
3
votes
1answer
50 views

If the sum of two independent random variables is $ L^{p} $, does it imply that each is $ L^{p} $?

Let $ X $ and $ Y $ be two independent random variables, i.e., $$ \forall a,b \in \Bbb{R}: \quad \textbf{Pr}(X < a,Y < b) = \textbf{Pr}(X < a) ~ \textbf{Pr}(Y < b). $$ Let $ p > 0 $ ...
1
vote
2answers
24 views

Calculate the variance of random variable $X$

A random variable has the cumulative distribution function $$F(x)=\begin{cases} 0 & x<1\\\frac{x^2-2x+2}{2}&x\in[1,2)\\1&x\geq2\end{cases}.$$ Calculate the variance of $X$. First I ...
0
votes
1answer
18 views

Determinate the joint probability marginal distribution

I have a random vector $(X,Y,Z,W)$ with the following Probability Mass Function: Which steps should I follow to determinate the joint probability marginal distribution of $(X,Y)$ in a similar table ...
2
votes
0answers
45 views

How to use Random Variables

I'm reading Introduction to Algorithms-Cormen, Leiserson, Rivest, Stein, Section 9.2 Selection in Expected linear time. Page-217 ...
-4
votes
2answers
67 views

The Random 666 number. [closed]

Does the number 666 more likely to come up more than other numbers like 777? The register, on stock exchange and at the pump? enter image description here enter image description here
2
votes
1answer
59 views

Why is it wrong to naively pick random points inside a disk

According to MathWorld, the naive way to randomly pick points inside a disk, by using two uniformly distributed variables that are polar parameters: $r \sim [0, 1]$ and $\theta \sim [0, 2\pi]$, is ...
3
votes
1answer
28 views

What assumptions did I make when I strengthened my independence criterion across a new random variable?

I have an algorithm which tries to calculate some $\operatorname{Pr(X | Y_1 Y_2 \dots )}$ (where juxtaposition means event intersection, "given $Y_1$ and $Y_2$ and ... have happened".) We have some ...
1
vote
1answer
27 views

$Z_n\rightarrow 0$ in probability and $W_n$ a series of random variables, implies $W_nZ_n\rightarrow 0$?

Yesterday you helped me prove that $Z_n\rightarrow 0$ in probability and $W$ a random variable, implies $WZ_n\rightarrow 0$ in probability. Does this imply that for every series $W_n$ of random ...
3
votes
5answers
91 views

How is $\min\{X,Y\}$ defined for $X, Y$ random variables?

How is $\min\{X,Y\}$ defined for $X, Y$ random variables? Is it defined as the $\min$ of their probability functions for a constant value they get? For example: $X\sim ...
1
vote
2answers
31 views

Convergence in $L^p$: $E[X 1_A] = E[X_n 1_A]$

Let $p>1$ and suppose that $X_n \rightarrow X$ in $L^p$ as $n \rightarrow \infty$. For $A \in \mathcal{F}_n=\sigma(X_0, \dotsc, X_n)$ it is written $$E[X 1_A] = E[X_n 1_A]$$ Can you explain me ...
2
votes
1answer
39 views

non-negative almost surely [closed]

I have a probability measure P and a non-negative sequence of random variables $(X_n)$ and the limit $X=\lim X_n$ exists P-almost surely. I would like to show that $X\ge0$ P-almost surely.
1
vote
2answers
33 views

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, proving $WZ_n\rightarrow 0$

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, I need to prove $WZ_n\rightarrow 0$. I was given a hint: show that for every $\delta,\epsilon<0$ we have ...
2
votes
1answer
28 views

Almost sure and $L_1$ convergence to different limits?

Let $X_n, n \geq 1$ be random variables. Suppose that $X_n \rightarrow 0$ $\mathbb P$-a.s. as $n \rightarrow \infty$. Moreover, suppose that $E[X_n]=1$. Now is it clear that there cannot be ...
3
votes
1answer
39 views

Expectation of random variable with domain $\mathbb{N} \cup \{ \infty \}$

Let us consider a random variable $X \colon \Omega \rightarrow \mathbb{N} \ \cup \{ \infty \} $. Is there a reasonable way to define the expectation value $E(X)$? How do I deal with the fact that $X$ ...
1
vote
1answer
34 views

Is $\sum_{x=1}^{\infty} P(X \le x, Y>x)$ not the same as $\sum_{x=2}^{\infty} P(X<x|Y=x)P(Y=x)$?

I'm trying to figure out how to calculate $P(X<Y)$ for discrete random variables, taking values in the positive integers (so excluding 0). I've come up with a few ways. In my consideration the two ...
1
vote
2answers
33 views

Expected value of coin flip sequence

I flip a fair coin independently 6 times. This gives me a sequence of heads and tails. For each consecutive 'HTH' in the sequence you win $5. So I define a random variable X to be the amount of ...
1
vote
3answers
54 views

Understanding random variables

I am just getting into probability theory, and don't quite understand the concept of random variables. It is presented to me as a transformation, it takes input from a background space of "events" and ...
1
vote
2answers
45 views

expectation of a nonlinear function of a Gaussian random variable [closed]

I am reading a book on Monte Carlo Simulation and I want to know where the formula blow come from. $$E\left[ X^4\right] {\text{ }} = 3\sigma _x^4 + 6\sigma _x^2\mu _x^2 + \mu _x^4$$ Suppose $X$ is a ...
1
vote
0answers
19 views

How to calculate the expectation $E[\sigma(1)\sigma(2)\sigma(3)]$, given a 2-wise hash function $\sigma: [d] \rightarrow \{-1,+1\}$

Given a 2-wise hash function $\sigma: [d] \rightarrow \{-1,+1\}$, i.e., $\sigma(a)$ will map any positive integer $a\leq d$ into real number $-1$ or $+1$ with equal probability. Then how to calculate ...
1
vote
1answer
31 views

Do we need to find an upper bound for the expectation of this stopping time?

From here: It looks like: It is supposed to say 'different from six' rather than 'different from three' $T = \inf\{m: X_{m} = X_{m+1} = X_{m+2} = 6\}$ In every triple $P(all \ 6) = 1/216$ ...
0
votes
2answers
42 views

Expected value of times we roll a pair of dice [duplicate]

I am trying to solve this using a geometric series. Pretty much the question goes like this: We roll a pair of fair dice repeatedly and independently and stop as soon as the sum of the dice is equal ...
3
votes
0answers
126 views

Random Variable: Ordered List of ints.

You are given an ordered list of integers : 1, 2, ...100. You then randomly permute (reorder) the integers. a.) Define a random variable that indicates whether or not a pair of integers in the list ...
1
vote
1answer
51 views

Probability that sum of two random variables is $1$

Let $X$ and $Y$ be two independent random variables with density functions $f_x(x) = x\exp\left(-\frac{1}{2}x^2\right) \mathbb{}$ where $x \in \mathbb{R}^+$ and $f_y(y) = \frac{1}{2}$ with $y \in ...
0
votes
1answer
39 views

Expected Value Of Dice Rolling Game

I am having trouble figuring out the expected value in situations were the examples are going to infinity. Example: I have a fair dice with 8 sides. I keep a counter ($k$) of the rounds I play. Each ...
1
vote
2answers
37 views

Expected Value Of Coin Flip Game

I am having a hard time getting comfortable with random variables and expected values. Here is the question: I flip two fair and indepedent coins. If the first coin comes up tails you loose $\$1$ ...
2
votes
0answers
38 views

Sum of inverse chi squared random variables

Let $X$ and $Y$ be two i.i.d. random variables that follow an inverse chi squared distribution. Let $\nu$ be the corresponding degrees of freedom parameter. What is the distribution of the sum ...
2
votes
2answers
31 views

A puzzle on random variables that must have the same sum

$X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ are random variables that take values in $\{0,1\}$. Their distribution is unknown, each variable may have a different distribution and they might be dependent. The ...
2
votes
1answer
78 views

Conditional pdf of a random variable that is a function of other random variables

Given a pair of random variables $X,Y$, the conditional pdf of $Y$ given $X=x$ is defined by $$f(y\mid x) = \frac{f(x,y)}{f_X(x)}$$ Now, suppose $Z$ is another random variable and $Y=g(X,Z)$. Then ...
1
vote
0answers
15 views

Confusion about the concept of ergodicity in random process

This is a basic question I came across when I started learning random processes. Suppose I have a random variable $f(t) = G(t) + n(t)$ where $G(t) = A\exp(-t^2/T^2)$ (i.e. a deterministic function ...
1
vote
1answer
34 views

Understanding the meaning of seed in generating random values?

I am working on a project and I'm reading this description on generating random bits in a file. They use the word "seed" in it. I've read what seed does, but I'm not quite sure how to apply it in this ...
2
votes
1answer
21 views

Estimate $m$ using method of maximum likelihood.

Estimate $m$ using method of maximum likelihood. In the box there are $91$ balls, where $m$ are red, and the rest are blue. To estimate unknown parameter $m$, at once $19$ balls are drawn, $7$ being ...
1
vote
0answers
42 views

Probability: finding the expectation of “overlapping events”

Suppose there are 666 coins with 6 different colors in a non-transparent box. 111 of them are white coins. 111 of them are black coins. 111 of them are yellow coins. 111 of them are red coins. 111 of ...
1
vote
0answers
19 views

Implications of convergence in mean square

Consider the sequence of real-valued random variables $\{X_n\}_n$. Suppose $\{X_n\}_n$ converges in quadratic mean to the random variable $X$. All random variables are defined on the same probability ...