Exercise 1.82

tsim=10;
λ[t_]=1/(t^2+1);
path=RandomFunction[InhomogeneousPoissonProcess[λ[t],t],{0,tsim}];
f=path["PathFunction"];
jumps=path["Times"];

Plot[f[t],{t,0,tsim},Exclusions->jumps,ExclusionsStyle->Dashed,AxesLabel->{"t","N(t)"},PlotStyle->Black]

𝒫=InhomogeneousPoissonProcess[λ[t],t];
Mean[𝒫[t]]
PDF[PoissonDistribution[Mean[𝒫[(Sqrt[2])-1]]],2]*CDF[PoissonDistribution[Mean[𝒫[1]]-Mean[𝒫[(Sqrt[2])-1]]],1]//N