Colorblindness is caused by a recessive gene on the X chromosome. Since men have only one X chromosome, if a man carries the colorblindness allele (gene form), he will have the trait. Women have two X ...

Abstract: In this chapter, we introduce the concept of a random variable and develop the procedures for characterizing random variables, including the cumulative distribution function, as well as the ...

In a raffle with 20 tickets, 6 tickets are drawn for prizes. The first prize winner gets $\$20$, 2 second prize winners get $\$10$, and three third prize winners get $\$5$. What is the sample space ...

In data science, we often deal with data that is affected by chance in some way: the data comes from a random sample, the data is affected by measurement error, or the data measures some outcome that ...

Abstract: This book introduces the basic concepts of set theory, measure theory, the axiomatic theory of probability, random variables and multidimensional random variables, functions of random ...

$\def \bb#1{{ \mathbb{#1} }}$ $\def \P#1{{ \mathbb{P} \left(#1\right) }}$ $\def \E#1{{ \mathbb{E} \left(#1\right) }}$ $\def \Var#1{{ \text{Var} \left(#1\right ...

The main property of a discrete joint probability distribution can be stated as the sum of all non-zero probabilities is 1. The next line shows this as a formula. The marginal distribution of X can be ...

College of Mathematics and Physics, Shijiazhuang University of Economics, Shijiazhuang, China. Science College, Beijing University of Civil Engineering and Architecture, Beijing, China. [1] W. Liu, ...