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

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0
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2answers
24 views

Calculate the Covariance of random variables that distribute normally

$X_1$ and $X_2$ are two independent random variables that distribute normally with mean $μ$ and variance $σ^2$. $Y_1 = X_1 + 2X_2$ $Y_2 = X_1 - 2X_2$ Calculate $Cov(Y_1,Y_2)$. Well, I ...
2
votes
2answers
40 views

How to calculate probability distribution of a function of two independent Poisson random variables?

I can't figure out how to determine the probability distribution function of $$aX + bY,$$ where $X$ and $Y$ are independent Poisson random variable. Basically, I want to check whether $aX+ bY$ ...
1
vote
3answers
40 views

Distribution of ages of 3 children in a family

Please consider the following problem: A family has 3 children, creatively named A,B, and C. (a) Discuss intuitively (but clearly) whether the event “A is older than B” is independent of the event “...
2
votes
1answer
36 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
4
votes
3answers
77 views

Series of independent Bernoulli variables

Let $X_1, X_2, \ldots$ be independent, identically distributed random variables with distribution $\text{Ber}(\frac{1}{2})$. Define the random varible: $$Y:=\sum_{n=1}^\infty \frac{X_n}{2^n}$$ ...
0
votes
0answers
30 views

In what sense does does linear dependence correspond to random variable dependence?

In linear algebra, there is a theorem that states that $\langle v, w \rangle = 0$ implies that $v$ and $w$ are linearly independent. Now let $V$ be a vector space of real-valued random variables on ...
0
votes
0answers
24 views

Is there a formula for the MGF of $Y=g(X)$?

Let $X$ be a real valued random variable with cumulative distribution function (CDF) $F_X$ and probability density function (DF) $f_X$. Suppose $g\colon\Bbb{R}\to\Bbb{R}$ is a differentiable, strictly ...
5
votes
3answers
552 views

Negative Variance

I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I have ...
2
votes
1answer
42 views

Probabilistic constraint implying deterministic constraint?

Suppose $X$ is an $N$-dimensional random variable $X := [X_1 \; X_2 \; \cdots \; X_N]$ such that all entries can either be 0 or 1 while satisfying the following: (i) $\mathbb{P}(X_i = 1) = p_i \; \; ,...
6
votes
2answers
78 views

Why does Slutsky's Theorem Fail to Generalize? [closed]

What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that $...
1
vote
0answers
9 views

Joint distribution of sum and summand

Let $Z_1$ and $Z_2$ be independent random variables with known distributions $F(.;\theta_1)$ and $F(.;\theta_2)$ of the same convolution closed family. Then $Y = Z_1 + Z_2$ has distribution $F(.;\...
13
votes
5answers
2k views

Are we guaranteed that the harmonic series minus infinite random terms always converge?

Consider the known harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows $$\sum_{n=1}^\infty a_n\frac{1}{n}$$ where $$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$ i.e. ...
1
vote
1answer
33 views

Finding the pdf of the difference of minimum and maximum of a finite set of random variables.

Let $X_i$ $ (1\leq i\leq n)$ be identically distributed uniformly on $(0,1)$. Let $U = \min_i(X_i)$, $V = \max_i(X_i)$. Find the pdf of $V-U$ This is what I did. I found the cdf and differentiated. ...
1
vote
2answers
27 views

Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$ Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases} $$ ...
0
votes
0answers
25 views

How to compute integrals using any probability law with Monte Carlo?

I am intrested in providing an estimation of : $\iint C(x,y)dP_X(x)dP_Y(y)$ I am able to generate random numbers from the distribution of $P_Y$ and $P_X$. Therefore I generate a big number (n=10 000)...
3
votes
2answers
57 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
3
votes
1answer
31 views

$\limsup$ sequence independent $\mathcal{N}(0,\sigma^2)$

In my lecture notes there is the following application of Borel-Cantelli's 2nd lemma: Let $(X_n)_{n\geq 1}$ be a sequence of independent $\mathcal{N}(0,\sigma^2)$-distributed random variables, with $\...
3
votes
1answer
31 views

Conditional distributions allowed pdf to take on single value?

My question is about Conditional probability distributions. From what I have learned, PDF's aren't allowed to take on singular values, yet I find that this definition seems to go out the window when ...
0
votes
1answer
27 views

Linearly Dependent Random Variables

Intuitively, what is meant to be captured by the notion of linearly dependent (real-valued) random-variables?
0
votes
0answers
29 views

What is the Standard Way to Define a Norm on a Vector Space of Real-Valued Random Variables?

Let $V$ be the vector space of real-valued random variables over $\mathbb{R}$. How does one traditionally define the norm on $V$?
0
votes
1answer
16 views

Intuitive definition of scaling random variables by a constant?

From how I understand scaling discrete random variables, we are multiplying all members in the set by the scaling constant. I.E if our random variable X = {1,2,3,4} and our scaling factor is $\alpha ...
1
vote
1answer
23 views

Scaling Random variables by a constant

First, let me state the problem: "Customers at Fred's cafe win a 100 dollar prize if their cash receipts show a star on each of the five consecutive days Monday...Friday in any one week. The cash ...
0
votes
0answers
71 views

Does this sequence of random variables converge almost surely?

I was trying to understand why almost sure convergence doesn't imply convergence of the mean and I encountered this answer. However, I do not understand why this sequence of random variables ...
1
vote
0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
5
votes
2answers
74 views

Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
1
vote
1answer
38 views

Convolution of PDFs is a PDF

Suppose $f$ and $g$ are PDFs of real-valued random variables. Show that the convolution $f\ast g$ of $f$ and $g$ (defined below) is also a PDF. $$(f\ast g)(x)=\int_{-\infty}^\infty f(y)g(x-y)\,dy.$$ ...
0
votes
1answer
49 views

How to simulate a delta-correlated random process

I'm trying to do the simulation described in the paper attached, but there is something I don't understand. The author says that the random variables which satisfy the relation (Eq. (4) in the paper) ...
0
votes
1answer
39 views

Probability Density Function of Random Variable which is Sum of other Random Variables

Let $X_0, X_1, X_2, ..., X_n$ each be non-identical independent random variables. Let $x_0, x_1, ... , x_n$ be possible values of each of those random variables. Let $\newcommand{\Pdf}{\...
0
votes
1answer
33 views

Probability Density Function of Random Variable which is Max of other Random Variables

Let $X_0, X_1, X_2, ..., X_n$ each be non-identical independent random variables. Let $x_0, x_1, ... , x_n$ be possible values of each of those random variables. Let $Pdf_{x0}(x_{0}), Pdf_{x1}(x_{...
0
votes
0answers
20 views

Showing independence of two independent identically distributed random variables.

The title might be confusing, here is the task: $X$ and $Y$ are two independent and identically distributed random variables with $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=0.5$ and $Z=X\cdot Y$. Show that $...
1
vote
0answers
37 views

Given the joint pdf of $X$ and $Y$, find the joint pdf of $W = X+Y$ and $T = X-3Y$.

Let the two-dimensional random variable $(X, Y)$ be whose joint probability function is: $f(x,y) = 1/4$, for $0\le x\le2$ and $0\leq y\leq 2$ a) Calculate the joint probability density ...
0
votes
1answer
33 views

Calculating PDF from Autocorrelation

I have a statement like this; A zero mean Gaussian random process $X(t)$ is wide sense stationary with the auto-correlation function $R_x(\tau) = 4e^{-2|\tau|}$ And I want to find the ...
1
vote
1answer
20 views

Tight sequence of rv's such that $V(X_n) \rightarrow +\infty$.

Let $(X_n)$ be a tight sequence of real valued rv's, i.e. $\displaystyle \lim_K\sup_n P\left(\left|X_n\right|>K\right)=0$, defined on a common probability space, such that $E\left(X_n^2\right)<+\...
0
votes
1answer
37 views

Probability of failure of individual components in parallel and sequential system

Not sure If I'm calculating these probabilities correctly was wondering if someone could lead me in the right direction otherwise. Probability of a channel working properly is 0.8 and all channels are ...
4
votes
1answer
76 views

Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables?

I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$...
2
votes
0answers
68 views

Coin-toss game with \$1 entry fee and \$3 payout [closed]

Imagine a coin-tossing bet game. You pay \$1 to play the game (for one toss), and if you win you get a prize of \$3. The \$1 to play is not refunded. The probability of winning and losing is equal. If ...
2
votes
3answers
66 views

Finding the density for $\min\{X, Y\}$

Problem: Let $X$ and $Y$ be independent and suppose that each has a $\text{Uniform}(0,1)$ distribution. Let $Z = \min\{X, Y\}$. Find the density $f_Z(z)$ for $Z$. Hint: It might be easier to first ...
0
votes
0answers
16 views

Using of Ito formula

I have a task- $\alpha,\beta \in R$ and $N(t)=e^{\beta t}\cos(\alpha W(t)).$ It is necessary to calculate $E[\cos(\alpha Z)]$, where Z is a standard normal random variable. I know that it is ...
0
votes
1answer
26 views

Distribution table for number of white balls without replacement

I have a box with $4$ white and $2$ black balls. I want to find the random variable distribution table for the number of white balls when I take $3$ balls without replacement. Now if we had ...
2
votes
0answers
166 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
1
vote
1answer
34 views

Source of faulty reasoning in expectation of product of random variables?

For iid $\xi_i>0$, with $\mathbb E[\xi_i]=1$, what is $\mathbb E[\prod_i^M\xi_i]$? Approach 1: $\mathbb E[\prod_i^M\xi_i]=\prod_i^M\mathbb E[\xi_i]=1$. There is another approach for $M\gg1$ with ...
1
vote
0answers
20 views

Generating Failure rate function using Octave.

I'm not familiarized with programming using mathematics. My problem is described below. The lifetime T of a device has pdf Find the failure rate function. I have learned that Failure rate ...
0
votes
0answers
27 views

Conditional probability and its expectation of continuous random variables

I have two subtle questions on the definition of conditional probability and its expectation. Here's the thing. Question 1. Definition of a conditional probability density function $f_{X|Y}(x,y)=\...
0
votes
1answer
19 views

Most likely value of negative binomial random variable

If $X$ is a negative binomial random variable let's say with $p =0.2$ and $r = 4$ then how can we calculate most likely value of $X$? I thought it is expected value but that is $20$ and I guess most ...
0
votes
0answers
17 views

Does $ P(X=x|W=\tilde{w}, P(Y=y|W=w)=a) $ make sense?

I have the following question. Consider a discrete random variable $X:\Omega\rightarrow \mathcal{X}\subset \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Consider other two ...
0
votes
1answer
18 views

Maximal distance between random variables in interval

Suppose there are two random variables $A, B$ constrained in the intervals $A \in [a_{min}, a_{max}]$ and $B \in [b_{min}, b_{max}]$. $a_{min}, b_{min} \leq 0$ and $a_{max}, b_{max} \geq 0$. I ...
0
votes
1answer
48 views

Proof of linearity for expectation given random variables are dependent

The proof of linearity for expectation given random variables are independent is intuitive. What is the proof given there they are dependent? Formally, $$ E(X+Y)=E(X)+E(Y)$$ where $X$ and $Y$ are ...
1
vote
1answer
25 views

Finding the distribution of a random vector in a conditional probability problem [closed]

Players A and B are playing a game of drawing coins from two boxes without returning/replacing them. Box1 has three coins with values 0, 1 and 2 and Box2 has two coins with values 1 and 2. In the game,...
0
votes
0answers
34 views

existence of some variance over chebyshev's inequality

From the basic knowledge I have, there must exist the variance for some epsilon greater than zero but less than 1 for which the Chebyshev's inequality holds. Now the scénario I have to verify using ...
0
votes
1answer
36 views

Proof of Chebyshev's inequality for a geometric random variable

I have learnt the Chebyshev's inequality for a continuous case like log-normal and normal distributions and in trying to understand the application I came across the question: For a geometric ...