3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
1
vote
1answer
22 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
0
votes
0answers
37 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
1
vote
0answers
26 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
1
vote
1answer
42 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
1
vote
3answers
62 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
1
vote
1answer
45 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
3
votes
4answers
143 views

Does this sum converge or diverge?

Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal ...
1
vote
1answer
35 views

probility, placing balls, covariance

Can you please help to see where I did wrong? There are 10 balls, and each ball to be place in bin 1 and bin 2. Each ball is placed indepedently. Let X be the number of balls in bin 1 and Y be the ...
2
votes
1answer
31 views

probability, transformation on Random Variable

This is a more general question about the transformation of a random variable. Say X is given as a certain distribution, and Y=g(X). If it asks to compute the pdf of Y, I am having trouble to ...
0
votes
0answers
55 views

probability, birthday paradox: need help to understand the solution

I need help to understand the following solution to a birthday paradox problem. problem:So you have $20$ people. Then let $P=$ # of pairs that share the common birthday. Compute ${\bf E}[P]$, ...
0
votes
1answer
21 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
22 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
22 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 ...
1
vote
1answer
34 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 ...
1
vote
1answer
85 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
35 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 ${\bf E}[XY]$. ...
0
votes
2answers
37 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 ...
1
vote
1answer
46 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
56 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 ...
0
votes
1answer
43 views

Given the distribution of a random variable $R$, who do you get a uniform random variable $U$?

Let us say you have a random variable $R$. How would one generate a uniform random variable $U$, with the maximum possible entropy (or infinite entropy, if $R$ has such)? (For simplicity, you may ...
0
votes
1answer
25 views

Density on the square, expected value

Let $f: [0,1]^2 \rightarrow \Bbb R^{+}$ a density function on the square. I suppose that the random variable $X=(X_1,X_2)$ has the density f with respect to the lebesgue measure. I denote ...
0
votes
1answer
33 views

Moments of Geometric Random Variable

Let $X$ be a geometric random variable i.e. it represents the number of consecutive failures before you get the first success where the success probability is $\rho$. We know $E[X] = 1/\rho$ and ...
5
votes
1answer
187 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
1answer
101 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
1
vote
2answers
19 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
1
vote
2answers
57 views

the maximum of two random variable

The maximum of two random varibles $X$ and $Y$ is: $$Z=\max\{X,Y\}= \begin{cases} X & \text{if } X \geq Y \\ Y & \text{if } Y \geq X \end{cases}$$ I don't understand. So if I roll two dice, ...
0
votes
0answers
18 views

How to generate normally distributed random numbers? [duplicate]

Is there any function that can generate normally distributed random numbers?
0
votes
1answer
68 views

random walk with dependent increment

Consider the following sort of random walk. The position of the walker at time $t$ is represented by the random variable $r(t)$, with $r(0) = 0$. The variable satisfies the following equation, $$ ...
5
votes
0answers
171 views

expectation value for minimum distance between random variables

note: edited to clarify boundary issue Suppose $x_i$, $i=1\dots n$, are randomly drawn from a uniform circular distribution between 0 and 1 (using periodic boundaries). Let $d_i$ be the distance ...
2
votes
0answers
67 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
1
vote
1answer
146 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
1
vote
1answer
283 views

problem on random variable in probability

A game consists of first rolling an ordinary 6-sided die once and then tossing a fair coin once. The score, which consist of adding the number of spots showing on the die to the number of heads ...
2
votes
1answer
96 views

nonlinear transform of Gaussian random variable that preserves Gaussianity

I recently know that following results. suppose that $x_1, x_2, x_3$ are independent real Gaussian random variables with $\mathcal{N}(0, 1)$. Then $$ \frac{x_1 + x_2 x_3}{\sqrt{1+x_3^2}} \sim ...
0
votes
1answer
106 views

Probability of Specific event occuring between 2 events?

Forgive me beforehand for what may be a question with an obvious seolution, but I havent had statistics courses in quite some time. I have an Excel File of approximately 3000 Events, each event has a ...
2
votes
2answers
55 views

Manipulating random decimals

I've been slowly working my way into the world of AI and its representations of random. Well as you can guess this occurs from a call to random() resulting in a ...
5
votes
1answer
147 views

A consequence of the law of large numbers

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, ...
0
votes
1answer
60 views

Sum of poisson random variables on a lattice

Consider a lattice $\mathbb{Z_+}$ and immagine that on each site $i \in \mathbb{Z}_+$ there is a number of particles $X_i$, where $X_i$ are i.i.d. Poisson random variables having expectation $\mu$. ...
2
votes
1answer
81 views

law of large number modified statement

The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$, $$ \forall ...
0
votes
1answer
85 views

Chernoff type Sum of independent random variables having exponential tails

Say I have n independent variables $\{X_1,X_2 \dots X_n\}$ with Expectation 0 such that $Pr(|X_n| > \alpha) < e^{-\lambda \alpha}$. Can we produce chernoff type inequalities for the sum of these ...
0
votes
0answers
47 views

Property of a random distribution.

I have to get an integral in a previous post Help on an integral.. Some guru gave me hints of how to approximate its value, but I need approximation with largely varying $a$ and $b$. I realized that ...
1
vote
1answer
28 views

Concentration of measure for sum of products

Let $X_i$, $i \in \{1 \ldots n \}$ be independent random variables taking value +1 or -1. I know that there exists a lot of results which talk about the concentration of $S_n = \sum\limits_i X_i$. Can ...
1
vote
2answers
89 views

What exactly does this physically mean?

Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the ...
1
vote
1answer
78 views

probability: random variable

From the Ross book ex.13 chapter 4: A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to sale with probability $0.3$ and the second will lead ...
1
vote
1answer
158 views

Rademacher random variables in terms of Bernoulli

I've found out that Rademacher random variables and Bernoulli random variables plays an important role in Probability theory. I am wondering how they are connected. For example, Let $r_i, i=1, ...
0
votes
0answers
31 views

Generate Constrained Vector of Random Numbers?

I'm having trouble creating a random vector $\vec{V}$ starting with a standard 0:1 randon number generator subject to the following set of constraints: (given parameters $D$, $L$, and $\theta$) The ...
0
votes
1answer
57 views

Expected Value of Exponential with IID Uniform Argument

Let us assume that we generate a discrete random variable $I \in \{-1,1\}$. Then, we know that $E\{I\} = \frac{1}{2} \cdot 1 - \frac{1}{2} \cdot 1 = 0 $. However, what is $E\{e^{jI}\}$? Where $j$ ...
0
votes
2answers
140 views

Deck of cards random variable

How would I compute the expected value of the number of kings randomly drawn pair of cards from a single deck? I tried $0: (\frac{48}{52})\cdot(\frac{47}{52})$ $1: (\frac{4}{52})\cdot ...
0
votes
1answer
1k views

Cumulative distribution function determine the random variable

I don't know that determine is the right word, but I try to explain. What I need to understand. :) So.. We know's that if a function fit this conditions: Monotonically non-decreasing for each of its ...
5
votes
2answers
512 views

Summing (0,1) uniform random variables up to 1 [duplicate]

Possible Duplicate: choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries? So I'm reading a book about ...