E ( X ) = d M X d t ∣ t = 0 = ( 3 + 64 t ) e 3 t + 32 t 2 ∣ t = 0 = 3 E(X)=\frac{dM_X}{dt}|_{t=0}=(3+64t)e^{3t+32t^2}|_{t=0}=3 E ( X ) = d t d M X ∣ t = 0 = ( 3 + 64 t ) e 3 t + 32 t 2 ∣ t = 0 = 3
E ( X 2 ) = d 2 M X d t 2 ∣ t = 0 = 64 e 3 t + 32 t 2 + ( 3 + 64 t ) 2 e 3 t + 32 t 2 ∣ t = 0 = 67 E(X^2)=\frac{d^2M_X}{dt^2}|_{t=0}=64e^{3t+32t^2}+(3+64t)^2e^{3t+32t^2}|_{t=0}=67 E ( X 2 ) = d t 2 d 2 M X ∣ t = 0 = 64 e 3 t + 32 t 2 + ( 3 + 64 t ) 2 e 3 t + 32 t 2 ∣ t = 0 = 67
σ = V ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = 67 − 9 = 7.6 \sigma=\sqrt{V(X)}=\sqrt{E(X^2)-(E(X))^2}=\sqrt{67-9}=7.6 σ = V ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = 67 − 9 = 7.6
P ( X ≥ a ) = e − a t M X ( t ) P(X\ge a)=e^{-at}M_X(t) P ( X ≥ a ) = e − a t M X ( t )
P ( X ≥ 3 ) = e − 3 t e 3 t + 32 t 2 = e 32 t 2 P(X\ge 3)=e^{-3t}e^{3t+32t^2}=e^{32t^2} P ( X ≥ 3 ) = e − 3 t e 3 t + 32 t 2 = e 32 t 2
P ( X < 3 ) = 1 − P ( X ≥ 3 ) = 1 − e 32 t 2 P(X<3)=1-P(X\ge 3)=1-e^{32t^2} P ( X < 3 ) = 1 − P ( X ≥ 3 ) = 1 − e 32 t 2
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