Probability theory is like the crystal ball of mathematics, helping us predict the likelihood of events occurring. The magic number always lies between 0 and 1, inclusive, where 0 means “impossible” ...

A discrete random variable is a type of random variable that can take on a countable set of distinct values. Common examples include the number of children in a family, the outcome of rolling a die, ...

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. \[\int_{-\infty}^{\infty}f\left(x\right)dx=1\] ...

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 ...

This program generates N amount of random variables for continuous uniform, exponential, Bernoulli, Poisson, Binomial, Gamma, and discrete distribution with the support set {1,2,3,4} and pmf P[W_i=j]= ...

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 ...

Abstract: Traditionally, the uncertainty qualification is utilized with the known probability distribution function (PDF). However, in some scenarios, the PDFs of some uncertain variables are modeled ...

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 ...