# Tagged Questions

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

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### 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....
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### 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 ...
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### 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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### 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 ...
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### 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 $x$...
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### Calculate the Joint Probability Density Function given X and Y.

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