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

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2
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2answers
30 views

how to understand the generation of cauchy distribution from uniform distribution?

I am learning some basic idea on generating cauchy distribution from uniform random generator $u \in [0, 1]$. I know it was discussed before in How to generate a Cauchy random variable, but during my ...
1
vote
1answer
26 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$ \frac{1}{n+1}\left[ \frac{1-\exp\left( \left(n+2 \right)it \right)}{ 1-\exp(it) } \right]$$ and taking its limit when ...
0
votes
0answers
14 views

Closed form expression for the S-transform of a random variable?

I'm trying to compute the S-transform as described in this review article on random matrix theory (section 2.2.6). They define it as $\Sigma_X(\gamma) = -\frac{\gamma+1}{\gamma}\eta_X^{-1}(1+\gamma)$ ...
-2
votes
1answer
35 views

Proving the variance of pareto random variable equals (a*lambda)/((a-1)^2*(a-2))

So my PDF for the Pareto distribution is: $$\dfrac{a\lambda^a}{x^{a+1}},\quad x\ge\lambda$$ To find the variance, you need to find the integral of $x^2\dfrac{a\lambda^a}{x^{a+1}}$ and subtract it from ...
1
vote
2answers
31 views

Solution to a covariance problem

Roll two dice, let $X$ be their sum, and $Y$ be the second roll subtracted from the first. Compute $\text{Cov}[X,Y]$. Brute-force calculating $\text{Cov}[X,Y]=E[XY]-E[X]E[Y]$ is fairly horrible, ...
0
votes
1answer
9 views

Proof that 2 geometric random variable is NB

can someone write me the proof of 2 geometric variable are negative binomial ? $X\sim G(p)$ and $Y\sim G(p)$ how can i proof that $Z=X+Y \sim NB(2,p)$?
2
votes
1answer
63 views

Conditional expectation, $X = \varphi (Y)$

Show that if $$\forall \omega \in A \ : \ X(\omega) = \varphi(Y(\omega)), \ \ A \in \Sigma_Y$$ (that is, the equality is true for $\omega \in A$), then $$\mathbb{E}(X|Y)(\omega) = \varphi(Y(\omega)) ...
1
vote
2answers
52 views

How to calculate the probability distribution function (PDF)?

Sorry for the dumb question, I've been struggling with understanding the probability distribution function formula, what does "x" and "d" stand for in the formula , and how to use the formula? I've ...
0
votes
0answers
7 views

how can find statistical and linear independence from sample space

Let sample space is $ S=\{\varsigma_1,\varsigma_2,\varsigma_3,\varsigma_4\} =\{-1, -0.5,0.5,1\}$ and define two random variables as $ X(\varsigma)=1/\varsigma $ and $Y(\varsigma)=2^{-\varsigma}$. a) ...
6
votes
1answer
39 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
1
vote
3answers
45 views

How to explain why the probability of a continuous random variable at a specific value is 0?

Consider X as a continuous random variable which can assume any value in [0, 1]. It is known that P(X=x)=0 where P is the probability density function. I want to understand this intuitively. The math ...
0
votes
1answer
40 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
0
votes
4answers
83 views

Expected value of the minimum of a non-negative random variable and a constant

X is a non-negative random variable. Define Y = MIN(X, c) where c is a constant. What is E[Y]? I am modeling the constant as another random variable whose pdf is Dirac Delta function: $f_{c}(x) := ...
2
votes
1answer
69 views

Sequence of finite, positive and i.i.d random variables and limit of $\frac{S_{n+1}}{S_{n}}$

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of finite, positive and i.i.d random variables and let's call $\mu:=E(X_{1})>0$ and $S_{n}:=\sum_{i=1}^{n}X_{i}$. We know that ...
0
votes
2answers
43 views

CDF of $-\ln X$ where $X$ is uniform on $(0,1)$

I'm having difficulty studying this part of the subject, because i can't get through this first example, can anyone help? Let $$X: U(0,1)$$ Find the distribution function of the following random ...
0
votes
0answers
17 views

In a group of 2n couples , in a random fashion 2n people are chosen.

In a group of 2n couples , in a random fashion 2n people are chosen. Find the distribution of the random variable X , which represents the number of couples in the chosen group. $\Omega$-probability ...
1
vote
1answer
28 views

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$.

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$. How can you do this? Do you need the change of variables technique? I am a bit confused about ...
1
vote
1answer
24 views

continuous probability density functions

Continuous distributions assign probability 0 to individual values. But, according to DeGroot, it doesn't mean that it is impossible for the random variable to take individual values. So, why not make ...
2
votes
2answers
54 views

Expectation of quotient of random variables

Let $X_1,...X_n$ be independent, identically and non negative random variables, and let $k\le n$. Compute: $$E\left[{\sum_{i=1}^k X_i\over \sum_{i=1}^n X_i}\right]$$ this question has already been ...
2
votes
1answer
24 views

Random number generator from a piecewise PDF

I'm trying to create a random number generator on the interval $(a,c)$ given a probability density function defined as: $$f(x) = \left\{ \begin{array}{lr} \dfrac{C}{x} &, x \in (a,b)\\ ...
2
votes
2answers
47 views

Is there a meaningful way to approximate a discrete random variable?

Is there a meaningful way to find a continuos approximation of a discrete random variable? Thoughts for the $L^2$ case If $X \in L^2$, then we may want to consider the subspace $V = C^1 \cap L^2$ ...
0
votes
1answer
33 views

Would it be safe to say that a random variable X is identially zero when its first and second moments are both zero?

Would it be safe to say that a random variable $X$ is identially zero when its first and second moments are both zero? If it is true, how would you prove this? This step is needed when we prove that ...
1
vote
1answer
45 views

How do we approximate sum of random variables?

Suppose we have independent, identically distributed random variables $X_n \notin L^1$. I would like to approximate, in some way, the distribution of their sum $\sum X_n$ .The problem is that these ...
0
votes
3answers
23 views

Equality in distribution

If $A\stackrel{d}{=}C$ and $B\stackrel{d}{=}D$, is it $A+B\stackrel{d}{=}C+D$, where $A,B,C,D$ are dependent random variables?
0
votes
2answers
25 views

Determining a mean and skewness of a probability density function

Is there any way to determine the mean and skewness of a probability density function WITHOUT integrating? I have the following function: $$ f(x)= \begin{cases} x&\text{if}\ 0\le x\le1,\\ ...
-1
votes
2answers
47 views

Ito Differential Equation example [closed]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
1
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0answers
38 views

Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
1
vote
1answer
31 views

Calculating the distribution of a compound random variable

Given $X\sim U(1, 0)$ and $Y\sim Exp(1)$, determine the density function of $Z:=\frac{X}{Y}$. Now, without looking up how to do it I tried to figure it out myself. The value of the density function ...
0
votes
1answer
18 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
1
vote
1answer
15 views

Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
0
votes
0answers
45 views

Game of Keno from Sheldon Ross Chapter 4

I am facing with the following problem: A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 ...
0
votes
0answers
24 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
0
votes
0answers
22 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
0
votes
1answer
69 views

Mean return time in Markov chain

Given the following Markov chain: $p_{0,1}=1$ (if we are in state 0, we must go to state 1) $p_{i,i+1}=p_{i,i-1}=0.5$ There are infinitely (countably) many states. I assume that $X_0=0$ and define ...
1
vote
1answer
30 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
0
votes
1answer
53 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
1
vote
2answers
21 views

Random Variable being $F$-measurable

It is said the Random variable is $F$-measurable if $\{X\leq x\}$ is an element of $F$. Is $X$ not $F$measurable once it is not less than or equal to $1$ $x$ or only for all?
1
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2answers
61 views

Inverse of a mean, exponential distribution, expected value

Could you help me find the expected value of this random variable? Let $X_1, X_2, ... $ be independent identically exponentially distributed with parameter $\lambda$ random variables. What is the ...
1
vote
1answer
42 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
1
vote
1answer
56 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
0
votes
0answers
12 views

Distribution of $\sum n_i(U_i-U_{(1)})$

Let $U_i$ be independent random variables with pdf $f_i(x)$ ($i=1,\ldots,k$) where $$f_i(x)=\frac{n_i}{\sigma}\exp(-\frac{xn_i}{\sigma}), x>0$$ Let $n=\sum n_i$ and $U_{(1)}=\min U_i$. ...
0
votes
2answers
34 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
-1
votes
1answer
32 views

Random Variables. A Probability Question.

We have the folloing problem in my probability class, and I want to know if I have outlined it correctly. In a Kingdom there are $3$ prisoners; $A,B,$ and $C$. The king says that two are condemned to ...
0
votes
1answer
57 views

How can you find $P(\frac{X}{Y-X}<0)$ if $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$

Let the independent random variables $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$, I want to prove that $P(\dfrac{X}{Y-X}<0)=(p-1)^{2}(p+1)$. Do I need the joint probability mass function for ...
0
votes
2answers
49 views

To use or not Bernoulli trials

I was asked to model the following experiment: Consider the n-th toss of a fair coin, and the event $E$ = '$k$-th toss results in heads'. I find easier to model the experiment using n random ...
0
votes
0answers
17 views

Support for a linear combination or transformation of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
0
votes
1answer
15 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
1
vote
0answers
31 views

Distribution Function Of a Random Variable X - Question

This is a homework question pretty much but I do not understand how to approach it. The distribution function of the random variable X is given: F(X) = 0, x < 0 x/2, 0 <= x < 1 2/3, 1 ...
0
votes
1answer
46 views

How to calculate the probability distribution function (PDF) and the cumulative distribution function (CDF)?

Sorry I'm a novice to both functions and just didn't get a clue how to solve this problem (having been reading the theories for the whole day but still ...) The problem is: We have now two investment ...
2
votes
0answers
22 views

Max cut problem

I've just looked at the standard proof using the probabilistic method stating that the max cut problem has a lower bound of $|E|/2$ for any graph $G=(V,E)$. More specifically if $X$ is the random ...