First of all, continuous mathematical expectation means that when a random variable takes a value in an interval, its mathematical expectation can be calculated by integrating the functions in the interval. Discrete mathematical expectation means that when a random variable can only take finite or countable infinite values, its mathematical expectation can be calculated by adding the probabilities of all possible values.
Secondly, continuous mathematical expectation is usually used to analyze and model continuous random variables, such as normal distribution and exponential distribution. Discrete mathematical expectation is suitable for discrete random variables, such as binomial distribution and Poisson distribution.
In addition, continuous mathematical expectation and discrete mathematical expectation are also different in application. Continuous mathematical expectation is often used to describe stochastic processes in continuous time or continuous space, such as Brownian motion and heat conduction equation. Discrete mathematics is expected to be more applied to discrete events or random processes in discrete space, such as queuing theory and network flow model.
In a word, although continuous mathematical expectation and discrete mathematical expectation are important tools to describe the characteristics of random variables, their calculation methods and application scope are obviously different.