ABSTRACT
As discussed in Section 4.2, the distribution function of a random variable X is a function F from ( - ∞ , + ∞ ) $ (-\infty , + \infty ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_1.tif"/> to R $ \mathbf{R} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_2.tif"/> defined by F ( t ) = P ( X ≤ t ) . $ F(t) = P(X\le t). $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_3.tif"/> From the definition of F we deduced that it is nondecreasing, right continuous, and satisfies lim t → ∞ F ( t ) = 1 $ \lim _{t \rightarrow \infty }F(t) = 1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_4.tif"/> and lim t → - ∞ F ( t ) = 0 $ \lim _{t \rightarrow -\infty }F(t) = 0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_5.tif"/> . Furthermore, we showed that, for discrete random variables, distributions are step functions. We also proved that if X is a discrete random variable with set of possible values { x 1 , x 2 , … } , $ \{ x_{1},x_{2},\ldots \}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_6.tif"/> probability mass function p, and distribution function F, then F has jump discontinuities at x 1 , x 2 , … , $ x_{1},x_{2},\ldots , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_7.tif"/> where the magnitude of the jump at x i $ x_{i} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_8.tif"/> is p ( x i ) $ p(x_{i}) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_9.tif"/> and for x n - 1 ≤ t < x n , $ x_{n-1} \le t < x_{n}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math6_10.tif"/> F ( t ) = P ( X ≤ t ) = ∑ i = 1 n - 1 p ( x i ) . $$ \begin{aligned} F(t) = P(X\le t) = \sum _{i=1}^{n-1} p(x_{i}). \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/math6_1.tif"/>
