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

Figure 12

Plot of 𝑓₁(𝑧,𝑡) given by (128)-(129) in Example 22 at different values of 𝑡={0,0.1,0.2,…,2} in the case that 𝜂=(𝑍₀,𝐴)^𝑇∼𝑁(𝜇_𝜂,Σ_𝜂) where 𝜇_𝜂 and Σ_𝜂 are given by (127) with 𝜇_𝑍0=1, 𝜇_𝐴=0, 𝜎_𝑍0=0.1, 𝜎_𝐴=0.1, 𝜌_𝑍0,𝐴=0.5, 𝑡₀=0, and 𝑏=1.

(a)

(b)