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0, otherwise, and p(0) = p Then E(Y) = (1−p)p−1 and Var(Y) = (1−p)p−2 Both of the above are called the geometric distribution, and are related by Y = X −1 4 Poisson distribution with mean (and variance) λ With λ > 0 a constant, X has pmL {f (t) ± g(t)} = L {f (t)} ± L {g(t)} 3 L {c f (t)} = c L {f (t)}, for any constant c Properties 2 and 3 together means that the Laplace transform is linear 4 The derivative of Laplace transforms L {(−t)f (t)} = F ′(s) or, equivalently L {t f (t)} = −F ′(s) Example L {t 2} = − (L {t})′ = 2 3 3 1Therefore, Ly δc = e−cs (s 1)2 1 Impulse response solution Example Find the solution (impulse response at t = c) of the IVP y00 δc 2 y 0 δc 2 y δc = δ(t − c), y δc (0) = 0, y 0 δc (0) = 0, c ∈ R Solution Recall Ly δc = e−cs (s 1)2 1 Recall Lsin(t) = 1 s2 1, and Lf (s − c) = Lect f (t) 1 (s 1)2 1 = L Solved Use The Graph Of F To Sketch Each Graph Y 10 4 2 Chegg Com I.f.l.y chords guitar