Question

2. Using calculus, find the mean and variance of an exponential distribution with a probability density function of f(x)-Aet, L? (Give a proof What is A in terms of

Answer 1

for above to be vald: $\int_{0}^{\infty }$ f(x) dx =1

$\int_{0}^{\infty }$ f(x) dx=$\int_{0}^{\infty }$ Ae^-x/L dx =-ALe^-x/L |^$\infty$₀ =-AL*(0-1) =AL=1

A=1/L

hence mean =E(X)=$\int_{0}^{\infty }$ xf(x) dx =$\int_{0}^{\infty }$ (x/L)e^-x/L dx =-(xe^-x/L-Le^-x/L) |^$\infty$₀ =-(0-L) =L

E(X²)=$\int_{0}^{\infty }$ (x²/L)e^-x/L dx =-(x²e^-x/L-2xLe^-x/L-2L²e^-x/L) |^$\infty$₀ =-(0-2L²) =L²

therefore Variance =E(X²)-(E(X))² =2L²-L² =L²