1. First, determine the probability density function (PDF) of random variables. The probability density function represents the probability distribution of random variables at each value. For continuous random variables, the probability density function is continuous.
2. Determine the definition of expected value of random variables. For discrete random variables, expected value is the sum of all possible values and their corresponding probabilities. For continuous random variables, the expected value is the integral of the product of the random variable and its corresponding probability density function.
3. According to the definition of expected value, the expected value of continuous random variables is expressed as an integral. Specifically, if the random variable is x and its probability density function is f(x), then the expected value is E[X]=∫[a, b]f(x)dx, where a and b are the interval boundaries of the domain.
4. Conduct integral calculation. According to the concrete form of probability density function f(x), the appropriate integration method is selected for calculation. The common integration methods are definite integration and numerical integration.
5. The calculation result is the expected value of continuous random variables.
It should be noted that for complex continuous random variables or probability density functions, it may be necessary to use special mathematical tools or software for integral calculation. In addition, in practical application, we need to consider the concepts of independence of random variables and conditional expectation, and further analyze and calculate related mathematical expectations.