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

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0answers
42 views

Lebesgue measures probabilities of random variables [duplicate]

Suppose that (Ω, F, P) is Lebesgue measure on [0, 1]. Define random variables X, Y and Z by X(ω)=ω, Y(ω)=2ω+3 and Z(ω)=4ω+1. Compute P(Z > a) and P(X < b and Y < c) as functions of a, b, c ∈ R....
2
votes
2answers
38 views

joint exponential distribution range problem

So the question asks: Let $X, Y$ be random variables, with the following joint probability density function: $$f_{X,Y}(x,y) = \left\{ \begin{array}{ 1 l } kye^{-y} & \mbox{if $0≤|x|≤y$}\\ 0 &...
3
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5answers
86 views

Suppose $X, Y$ are random variables with the equal variance. Show that $X-Y$ and $X+Y$ are uncorrelated.

Suppose that $X$ and $Y$ are random variables with the equal variance. Show that $X-Y$ and $X+Y$ are uncorrelated. I get I should use the equation $$E[XY] = E[X]E[Y]$$ For the first part I get $$E[(...
1
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0answers
13 views

Multivariate distribution with the same kurtosis as normal distribution

Good morning. I am writing a thesis about testing multivariate normality. I would like to do a comparison of power of some tests against given alternatives based on Monte Carlo simulations. I have a ...
0
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1answer
26 views

Show $E(XD|Z)=E(X|Z, D=1)Pr(D=1|Z)$

Consider two real valued random variables $X,Z$ and a dummy variable $D$. Is it true that $$ E(XD|Z)=E(X|Z, D=1)Pr(D=1|Z) $$ ? Why?
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1answer
23 views

Independence of random variable and random vector implies conditional independence

I have this interesting problem from probability theory stating as follows: Let $ X, Y, Z $ be three random variables such that $ Z $ and the vector $ (X,Y) $ are independent. We are to prove that ...
0
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0answers
51 views

Density function of $\sqrt{(\mathcal{N}_1(\mu,\sigma^2))^2+(\mathcal{N}_2(\mu,\sigma^2))^2}$ (Random walk)

I have 2D random walk and I would like to find out what distance I will travel after 200 steps. So I introduce two random variables $Z^{(200)}_x$ and $Z^{(200)}_y$ which tell me probabilities of my $x$...
0
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1answer
42 views

Calculate the Joint Probability Density Function given X and Y.

Given \begin{equation} X = \sqrt{-2\ln(U_1)}\cos(2\pi U_2);\quad Y = \sqrt{-2\ln(U_1)}\sin(2\pi U_2)\end{equation} where random variables $U_1, U_2$ have the continuous uniform distribution $U(0,1)$ ...
0
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0answers
28 views

On Poisson point process, mapping and Bernoulli experiment

I have just begun to read about Poisson point processes (PPP), and I think I understand the general idea. I am very interested on the possible application of the mapping theorem for 1-D processes (...
1
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1answer
27 views

A positive square integrable random variable whit non square integrable inverse

I'm looking for an example of a Square Integrable Random Variable, whose multiplicative inverse is not Square Integrable.
3
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3answers
37 views

What is the mean value of $\max\limits_{j}X_j$?

Let $X_j$ be a random variable that is $1$ with probability $x_j^*$, and $0$ with probability $1-x_j^*$. The random variables $X_j$ are independent and $j$ belongs to $\{1,\ldots,n\}$ for some ...
0
votes
1answer
22 views

How to convert a continuous probability density function into discrete samples?

What are the different methods that can be used to convert a probability density function of a continuous random variable into a discrete random variable? I have studied the uniform width method but ...
1
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2answers
30 views

Independence and conditional independence equivalence in probability theory

I was recently given this in my probability theory class on the different meanings of independence: Let X,Y and Z be three random variables. We are asked the following questions, to prove or ...
1
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1answer
58 views

Proving that the entropy is zero given conditional entropies

Let's suppose we have 4 random variables $X,Y,Z$ and $T$ and that the following equations hold about the entropy: $$H(T|X)=H(T)$$ $$H(T|X,Y)=0$$ $$H(T|Y)=H(T)$$ $$H(Y|Z)=0$$ $$H(T|Z)=0$$ Also, the ...
0
votes
1answer
20 views

Conditional Probability of joint discrete random variables

I have screenshot the entire question, but I have only about (f). How do you find the probability of P(Z|X=3)? I am confused about that part. I understand how the expected value is calculated. I just ...
2
votes
2answers
116 views

expected number of cards drawn exactly once (with replacement) [closed]

Suppose there are $N$ cards, $1,2,\dots,N$. We start drawing cards (with replacement), until each card has been drawn at least once (we stop when the last card is drawn for the first time). Let $x$...
1
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2answers
36 views

Correlation of sum of independent variables with its parts. if Z=X+Y, what is Cor(Z,X)?

If $Z = X + Y$, where $X$ & $Y$ are independent random variables, is there some formula to work out $\rho(Z,X)$, based on $\sigma_X$, $\sigma_Y$? For example, I've noticed that for $\sigma_X$ = $...
0
votes
1answer
74 views

Joint PDF in a circle area.

I don't understand how can I solve this. My only guess it's that it's related with the probability of the circle area of c. The coordinates X and Y of a point are independent zero mean normal random ...
-2
votes
2answers
33 views

Finding the distribution of a random variable [closed]

Let $U$ be a uniform random variable on $[0,1]$. What is the distribution of $-\log(1-U)$? (Answer: $25\%$) Solution. \begin{align} \mathbb{P}(-\log(1-U)\le x) &=\mathbb{P}(1-U\ge e^{-x})...
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1answer
73 views

Gaussian random variables independent iff statement

Let $X_1, \ldots , X_n:\Omega \to \mathbb{R}$ be random variables such that $(X_1, \ldots , X_n)$ is a Gaussian vector in $\mathbb{R}^n$. I want to show that $X_1$ and $\sigma(X_2, \ldots , X_n)$ are ...
0
votes
2answers
38 views

10 Coins being tossed and calculating certain probabilities

Last lecture the professor addressed us with a problem that I I'v may misunderstood but I am certain it is one of the two . In both problems we have 10 regular (no biased) coins on a table , and we ...
1
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2answers
42 views

What version of the Central Limit Theorem is this?

In lectures the following was written down on the board $$\mathbb{P}\left(\frac{X_1+\dots + X_n}{\sqrt{n}}>X\right)$$ Obviously this is in an incomplete state, and I all I managed to catch ...
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0answers
38 views

Adequacy of Monte-Carlo simulations

Suppose we have a number of independent random variables of the form $X_1 \sim U[a_1,b_1], X_2 \sim U[a_2,b_2], X_3 \sim U[a_3,b_3]$. Now, suppose we generate a random variable $Y$ as follows: $$Y = \...
2
votes
2answers
95 views

Joint PDF of two random variables in a triangle

Let the random variables $X$ and $Y$ have a joint PDF which is uniform over the triangle with vertices at $(0, 0), (0, 1 )$ and $(1, 0)$. Find the joint PDF of $X$ and $Y$. So ...
0
votes
2answers
16 views

Help on discrete variables

I need help with interpreting sentences in discrete random variables and how to convert it to tables,the question goes "A fair die is thrown once.A random variable represents the score on the ...
1
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1answer
35 views

average time for two light bulbulbs with IID exponential RV

I have two identical lightbulbs, each is an IID exponential random variables with mean 1000 hours. How long does it take for the first bulb to burn out, on average? For the second to burn out, on ...
0
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2answers
51 views

What kind of distribution in this chart?

Could you tell me what kind of distribution is this? Chart This is the data: ...
0
votes
1answer
43 views

Finding probability mass function for Y= 2x+4

I'm studying discrete random variables and pmf. Everything seems to make sense except when I need to find the pmf for random variables like this. I'm given the following information. I understand ...
1
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1answer
43 views

Distribution of infinite nested radicals with random terms

Let's say we have an infinite nested radical with random terms (positive integers) which can take on finitely many values in the range $(n_1,n_2)$. What would the distribution look like for these ...
2
votes
3answers
47 views

Let $X = \dfrac{1}{25} \sum\limits_{i=1}^{25} X_i$ and $Y =\dfrac{5}{2}X - \dfrac{2}{5}$. What is $P(|Y| > 1)$?

Suppose that $X_1,X_2,\ldots,X_{25}$ are independent random variables from $\mathcal{N}(1, 4)$. Let $X = \dfrac{1}{25} \sum\limits_{i=1}^{25} X_i$ and $Y =\dfrac{5}{2}X - \dfrac{2}{5}$. What is the ...
0
votes
1answer
47 views

Characteristic function and cdf

How do we find the points of discontinuity of a distribution of a random variable if the characteristic function $\phi (t)$ is given? How to find the cdf of a random variable X if the characteristic ...
2
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0answers
62 views

When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
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2answers
24 views

Determine individual distribution when knowing joint distribution of $2$ random variables X and Y

I have the joint distribution of $2$ random variables $X$ and $Y$. Here: How can I determine the distribution of $X$ and the distribution of $Y$ knowing this? I have tried using the following ...
5
votes
2answers
78 views

Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...
1
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2answers
28 views

what is the distribution of $\sum_{i=1}^n a_i^2$, where $a_i$ is independent Gaussian variable with zero mean and $b_i$ variance

If all the variance of all the Gaussian variables are the same, saying 1, then the distribution is Chi-squared https://en.wikipedia.org/wiki/Chi-squared_distribution. In my question, the variance are ...
0
votes
0answers
27 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 \...
0
votes
1answer
46 views

What are the expected value and the standard deviation of the number of games the final will take?

In the final of the World Series Baseball, two teams play a series consisting of at most seven games until one of the two teams has won four games. Two unevenly matched teams are pitted against each ...
0
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1answer
28 views

Conditional joint density of $Z=Y/X$

So the question asks : Let X and Y be random variable with joint density: $f_{X,Y}(x,y) = c $ when $x^2 +y^2 ≤1 $ $f_{X,Y}(x,y) = 0 $ in other situations (a) Find the value of the constant $c$. (...
0
votes
0answers
28 views

Covariance of functions of Gaussian random Variables

The two random variables $x$ and $y$ are Gaussian distributed with some Covariance. f is a non-linear stationary function. Now the following algorithm is applied: $ \underline{Algorithm:} \begin{...
0
votes
1answer
31 views

Which is correct formula for Plot/Gibbs distribution?

Helllo all, I am using Bayesian rule to classification the data. It has two term: likelihood and prior terms. The prior term can estimate by Gibbs distribution. According to the MRF-GRF (page 12),book ...
1
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1answer
21 views

What is the distribution function of a random outcome in closed interval [0,1]

a point is thrown at random on the interval [0,1], and if the outcome is x, you get 100x dollars. Y represents the amount of money you get. What is the pdf of X? what is the pdf of Y? I thought the ...
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0answers
19 views

Preserving stationarity of a process

This question is just for personal understanding. Is there common knowledge out there of functions that preserve stationarity of a process? More concretely, suppose $x(t)$ is WSS (wide-sense ...
1
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1answer
44 views

Chebyshev inequality for some random variable [closed]

I am looking for some help with the chebyshev problem that I am not quite sure how to do and am stuck on. Could someone provide some advice for the following question? Let $Z_{1}$, $Z_{2}$, $Z_{3}$, ...
0
votes
2answers
45 views

Derivative of a random number

I would like to numerically solve a differential equation which contains a derivative of a random number (using a finite difference method with a time step $\Delta t$). Let say I need to solve for $y(...
0
votes
0answers
36 views

Linear approximation of a random vector (X,Y) in the form of Y = aX + b

Problem I have a random variable vector (X,Y) whose joint mass function is given by the below table (X = j, Y = k). Additionally, the correlation of the vector is given by : E[XY] = probability of ...
0
votes
1answer
30 views

What does $X\sim Pois(\lambda S)$ mean?

I'm wondering what exactly does $X\sim Pois(\lambda S)$ mean when $S$ is a random variable as well? I guess $\mathbb{P}(X=k)=\frac{(\lambda S)^k}{k!}e^{-\lambda S}$ but still I do not know what that ...
0
votes
0answers
15 views

Var(X) using the exponential distribution family for the Gamma distribution

I have written the Gamma distribution ($\frac{1}{\Gamma(\mu)}r^\mu x^{\mu - 1}e^{-rx}$) in the normal form for the exponential family of distributions which is given as follows; $$exp\{\eta_1 log(\...
0
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1answer
58 views

Conditional entropy under quantization

Let $X$ be a continuous random variable and $X^n$ its quantization that becomes finer with larger $n$. Let $Y$ be a deterministic function of $X$. Then we have that the conditional entropy $$H(Y|X) = ...
0
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0answers
28 views

Independent composed random variables

Let me ask if it is possible to prove the following. Let $X_i, \, i=1,\ldots,n$ iid random variables and $Y_i, \, i=1,\ldots,n$ iid random variables. Now we define the composed random variables $$ U = ...
1
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0answers
20 views

Probability of bike rack being filled given distributions of bikes leaving and returning

Say we have $m$ spaces in a bike rack, which can be accessed by $n>m$ specific bikes. A bike leaves the rack at a random time $t$ according to a distribution $P_l(t)$ and returns a time $t'$ after ...