# Tagged Questions

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

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### How to compute the sum of random variables of geometric distribution

Let $X_{i}$, $i=1,2,\dots, n$, be independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$. How to compute the PDF of their sum $\sum_{i=1}^{n}X_{i}$? I know ...
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### Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ...
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### If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
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### Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two sigma-...
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### Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
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### Difference of two binomial random variables

Could anyone guide me to a document where they derive the distribution of the difference between two binomial random variables. So $X \sim \mathrm{Bin}(n_1, p_1)$ and $Y \sim \mathrm{Bin}(n_2, p_2)$,...
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### Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and $\mathfrak{T}$...
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### sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
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### Generate Correlated Normal Random Variables

I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $X_1$ and $X_2$, from the standard normal distribution. Then from there make $X_3$ ...
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### Finding probability P(X<Y)

How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent random variables.
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### If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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### What does the value of a probability density function (PDF) at some x indicate?

I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$. Now, a probability density function of of a continuous random variable X ...
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### Fixed points in random permutation [closed]

Suppose two random permutations of the numbers 1 to n placed side by side. a) Calculate the expectation number of fixed points for $n = 5$. b) Find the value of expectation in the amount of fixed ...
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### The concept of random variable

I'm reading Bernt Oksendal's "Stochastic Differential Equations" (edition 6) and I got quite confused on the conceptions. Please kindly help. I don't understand what is an event in the definition of ...
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### Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $X_n \Rightarrow 0$ (convergence in distribution) if and only if $(\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1$, where $\phi_n(t)$ is ...
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### Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest ...
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### Correlation between three variables question

I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
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### Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
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### What is the chances of a duplicate in this equation

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...