# Tagged Questions

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### Using the Probability generating function to find the probability of ultimate extinction

I am having problems with an exam question from a past paper, help would be appreciated: Let $X_n$ be the number of carriers of a family name in the nth generation and suppose $X_0=a$. Suppose ...
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### Compute a probability using stochastic simulation

My problem is to solve the following optimisation problem using GA (Genetic Algorithm)and stochastic simulation. The goal is to solve the maximisation problem : \begin{equation*} \begin{aligned} ...
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### Finding the highest probability balls

I'd appreciate any insights or references to research regarding the following: Suppose you have a discrete metric space with a probability distribution on it. also, suppose that I'm given a number ...
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### Convergence in distribution and limiting distribution

I'm trying to solve the problem below: Let $X_1,...,X_n$ be independent with PDF $f(x)=e^{-x}$ if $x>0$ and zero otherwise and define $$X_{(n)} = \mathrm{max}\{X_1,..,X_n\}$$ Find the CDF of ...
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### What is the distribution of the trace of an inverse Wishart distribution?

Given a Wishart distributed matrix $S$, what is the distribution of $\text{Tr}(S^{-1})$? What I can get to is Each diagonal element of $S$ is inverse gamma I can get the distribution of a sum of ...
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### Looking for a mathematical book on probability and statistics

The Short question: Where can I find a book for the theory of probability and statistics that teaches from scratch in a rigorous (very important condition) way? The book must not be elementary, but it ...
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For bivariate zero-mean normal distribution $P(x_1,x_2)$, the quadrant probability is defined as $P(x_1>0,x_2>0)$ or $P(x_1<0,x_2<0)$ and according to ...
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### How to show that $\mathrm{mgf}$ $M(s)$ and $\mathrm{pgf}$ $P(s)$ are related?

Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $s∈(-s_0,s_0),s_0>0$. ...
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### Understanding multinomial distribution

There are K categories. Let x be a discrete random variable taking on values $1,2...,K$. Given data $\textbf{X}= \{x_1, x_2, ..., x_N\}$, and suppose $\textbf{p}$ is a vector probabilities where, ...
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### How to compute similarity between two Dirichlet distributions?

I have a set of distributions and I would like to compute their similarity. Each distribution is a distribution over distributions of topics. If I have just a distribution over some topics, I could ...
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### Difference of random variables subject to non-negative values

This is probably close to trivial for stats experts. Yet, although having searched the web for quite a while now, I have not been able to come up with a satisfactory answer. Given two (approximately) ...
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### Book recommendations for beginners about probability distributions

I'm studying machine learning and every book I open I bump into chi-squared distribution, gamma-function, t-distribution, Gaussian, etc. Every book I have opened so far only defines what the ...
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### The derivation of the Mahalanobis distance formula [closed]

I recently asked about the Mahalanobis distance and I got pretty good answers in this post: Bottom to top explanation of the Mahanalobis distance? I think I got the idea, but what I still felt ...
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### Explain probability measure and probability distribution

Dear statistics community, I seek your help to understand the jungle of concepts about probability theory. I am struggling to form a coherent understanding of terms like probability mass function, ...
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### Characteristic function and Fourier transform

I understand the definition of characteristic functions used in probability theory: For a random Variable $X$ with probability density function $f_X$ the characteristic function is defined as: ...
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### Conditional distribution of quadratic forms

Given that $Y$ follows multivariate normal distribution ,i.e, $N_n (0, \sigma^2 I_n)$, we want to find the distribution of $Y'Y$ given that $a'Y=0$ where $a$ is a non zero constant vector. I know ...
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### Sampling distributions help with questions [duplicate]

Possible Duplicate: Probability of mean of random sample being in a certain range When a pizza restaurant’s delivery process is operating effectively, pizzas are delivered in an average of ...