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.
Highlights
For a continuous random variable as X, X R, which meet the condition X 'X, obviously, there is a probability density function f x, meet the condition that 3x f f t dt Fx
L are A xm, g xm and B xn, g xn, if it is need to required out linear integral of L, there are
If to prove the existence of linear probability density function, that is to prove the existence of h x, it could be based on the equation of the probability like 3g x h u du
Summary
For a continuous random variable as X , X R , which meet the condition X 'X , obviously, there is a probability density function f x , meet the condition that. With reference to examples of curvilinear integral[1,2], If the integral domain of X , which equivalent to interval of the value of random variable X from real number axis, changes to the function image ( or a part of it ) of g x which is a continuous function or countable piecewise function, there should be another function h x in real number field, according to the uniqueness of probability, it meets. H x are called the linear probability density function of the continuous random variable X in the real number field. Its need to define the linear probability density function, and prove its existence and the number of the function in real number field
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