Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

Table 1

List of the thirteen different cases considered to perform the full study depending on whether the IVP is homogeneous (H) or non-homogeneous (NH) and the form that uncertainty is considered into the problem: just through one RV: 𝑍₀, 𝐵, or 𝐴 (Cases I-II-III.1, I-II-III.2, and III.3), two RVs: (𝑍₀,𝐵), (𝑍₀,𝐴), or (𝐵,𝐴) (Cases I-II.3 and III.4-5-6), and the three involved RVs: (𝑍₀,𝐵,𝐴) (Case III.7).

Type	IVP	Case
H	𝑍̇(𝑡)(𝑡)𝑍𝑡=𝐴𝑍0=𝑍0    (I)	Case I.1	𝑍0 is a random variable
		Case I.2	𝐴 is a random variable
		Case I.3	(𝑍0,𝐴) is a random vector
NH	𝑍̇(𝑡)𝑍𝑡=𝐵0=𝑍0    (II)	Case II.1	𝑍0 is a random variable
		Case II.2	𝐵 is a random variable
		Case II.3	(𝑍0,𝐵) is a random vector
	𝑍̇(𝑡)(𝑡)𝑍𝑡=𝐴𝑍+𝐵0=𝑍0  (III)	Case III.1	𝑍0 is a random variable
		Case III.2	𝐵 is a random variable
		Case III.3	𝐴 is a random variable
		Case III.4	(𝑍0,𝐵) is a random vector
		Case III.5	(𝑍0,𝐴) is a random vector
		Case III.6	(𝐵,𝐴) is a random vector
		Case III.7	(𝑍0,𝐵,𝐴) is a random vector