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

Figure 11

Plot of 𝑓₁(𝑧,𝑡) given by (113)-(114) in Example 21 at different values of 𝑡={0,0.25,0.5,…,2} in the case that 𝜂=(𝑍₀,𝐵)^𝑇∼𝑁(𝜇_𝜂,Σ_𝜂) where 𝜇_𝜂 and Σ_𝜂 are given by (112) with 𝜇_𝑍0=1, 𝜇_𝐵=0, 𝜎_𝑍0=0.1, 𝜎_𝐵=0.1, 𝜌_𝑍0𝐵=0.5, 𝑡₀=0, and 𝑎=−1.