So far, you learned about discrete random variables and how to calculate or visualize their distribution functions. In this lesson, you'll learn about continuous variables and probability density ...

So far, you learned about discrete random variables and how to calculate or visualize their distribution functions. In this lesson, you'll learn about continuous variables and probability density ...

Probability theory forms the mathematical backbone for quantifying uncertainty and random events, providing a rigorous language with which to describe both everyday phenomena and complex scientific ...

Random analytic functions are a fundamental object of study in modern complex analysis and probability theory. These functions, often defined through power series with random coefficients, exhibit ...

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

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, ...

Density functions are nonnegative for all real numbers but greater than zero only at a finite or countably infinite number of points. Density functions are nonnegative for all real numbers and are ...

The joint probability density function \(f\) of two random variables \(X\) and \(Y\) satisfies, for every \(a_1 b_1\) and \(a_2 b_2\), \[ P(a_1\le X\le b_1, a_2\le Y ...

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 ...