Theorem 1 The stochastic transformation of aswhere is a constant scalar parameter and is a univariate normally distributed scalar , has a cumulative distribution function and density function as below: Theorem 2 Proposition (1) – The probability function of is given by as follows: where . Proof: The cumulative distribution function of is given by (see Tellambura, 2008): The proof that the density function takes the form as given above can be shown by differentiating with respect to and using the last result from Lemma 1. Theorem 2 Proposition (2) – The moment generating function of is given by: , where , , , and . Proof: where . The moment generating function of is given by:
Theorem 1 The stochastic transformation of aswhere is a constant scalar parameter and is a univariate normally distributed scalar , has a cumulative distribution function and density function as below: Theorem 2 – Proposition (1) The probability function of is given by as follows: where . Proof: The cumulative distribution function of is given by (see Tellambura, 2008): The proof that the density function takes the form as given above can be shown by differentiating with respect to and using the last result from Lemma 1. Theorem 2 – Proposition (2) The moment generating function of is given by: , where , , , and . Proof: where . The moment generating function of is given by:
Count_Discrete_Paper_2013.docx