The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof

¹ College of Software, Tianjin University, Tianjin, China
² College of Mathematical Science, Capital Normal University, Beijing, China

Abstract

For a continuous random variable in real number field, there must be a distribution and also a probability density function of this random variable. If there is a known function with this random variable as independent variable, its image is a smooth or piecewise smooth line, there must be at least one function that takes this random variable as its independent variable, these functions are bounded on the image of the first function. Any one of these functions conduct line integral operation to the line segment or arc length of the certain image of the first known function is the cumulative probability of this continuous random variable interval corresponding to the section of the image for line integral operation. A general designation for these functions are linear probability density function of continuous random variables. Conduct line integral operation to the linear probability density function and conduct integral operation to the probability density function have same results of the cumulative probability of continuous random variable. By the way, Line integration including curve integration. According to the uniqueness of the probability, the existence and the number of linear probability density function can be proved and calculated.