The total area under the curve must equal 1, representing the fact that the probability of some outcome occurring within the entire range is certain. \[\int_{-\infty}^{\infty}f\left(x\right)dx=1\] ...

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 ...

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 ...

The Virginia Lottery offers a game called the New Year's Millionaire Raffle for which the top prize is one million dollars. There are 375,000 tickets sold, of which 508 are winners. There are three ...

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 ...

A **variable** is said to be **random** if its values are determined by a random experiment. In other word, **random variable** is a numerical description of the ...

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

A discrete random variable is a type of random variable that can only take on a countable number of distinct values. A discrete random variable is a variable whose possible values are distinct and ...