Summary: The content provided is a mathematical derivation showing that a mixed Poisson distribution with an inverse Gaussian mixing distribution is equivalent to a Poisson-ETNB distribution.

Summary: We will get into how to get different moments of different distributions and possible skewness and kurtosis of different distributions. We will also discuss the relationship between different moments and the moment generating function.

Summary: The given text consists of two parts. The first part contains two examples along with their proofs concerning inequalities. In the first example, it is proved that if 'a_i' are real numbers such that the sum of their squares, the summation of (a_i^2), is less than or equal to 4, then the sum of their cubes, the summation of (a_i^3), will be less than or equal to 8. The proof involves utilizing the AM-GM inequality to demonstrate that the sum of the cubes is bounded by 8. In the second example, an inequality is proved for non-negative numbers 'a', 'b', and 'c': a^3 + b^3 + c^3 - 3abc is greater than or equal to 2((b+c)/2 - a)^3.

Summary: The provided text discusses the concept of compound distributions, which arise when combining two independent distributions. The probability generating function (pgf) of a compound distribution is shown to be the composition of the pgfs of the primary and secondary distributions. Some of the examples are excerpted from Loss Model (Sixth Edition).

Summary: In this blog, We introduces various methods in point estimates and their definition. The blog follows from lecture note Point Estimate from professor Xiaojing Ye in Georgia State University