Let X be a r.v. represents the life time of lamps which is produced by a factory and has a p.d.f given by: f(x)=ce^(-2x) , x>0 The standard deviation of the life time of lamps

Since f(x) is p.d.f,    then ∫_(-∞)^∞ 〖f(x)〗 dx=1
∴1=c∫_0^∞ e^(-2x)  dx=c[-1/2 e^(-2x) ]_0^∞=1/2 c  →c=2
 E(X)=∫_(-∞)^∞ x  f(x)dx=2∫_0^∞ x e^(-2x)  dx=
2[-1/2 xe^(-2x) ]_0^∞+1/2 ∫_0^∞ e^(-2x)  dx=1/2 e^(-2x)
 E(X^2 )=2 ∫_(-∞)^∞ 〖x^2 f(x)dx〗=2∫_0^∞〖x^2 e^(-2x)  dx〗=
2[-1/2 x^2 e^(-2x) ]_0^∞+2∫_0^∞ 〖xe^(-2x) 〗 dx=1/2 e^(-2x)
∴V(X)=E(X^2 )-(E(X)) ̅^2
=1/2 e^(-2x)-[1/2 e^(-2x)  dx]^2=1/2-1/4=1/4
∴σ_x=√(V(X))=√(1⁄4)=1/2