Question

The lifetime of new engines T years has a continuous pdf f(t)={d/t² if x>or equal1

{0,elsewhere

the value of the constant d,hence determine the mean and standard deviation T

Answer 1

$$\displaystyle\int_{-\infin}^{\infin}f(t)dt=\displaystyle\int_{1}^{\infin}\dfrac{d}{t^2}dt$$

$$=\lim\limits_{x\to \infin}[-\dfrac{d}{t}]\begin{matrix} x \\ 1 \end{matrix}=d=1$$

$$k=1$$

$$mean=\mu=E[T]=\displaystyle\int_{-\infin}^{\infin}tf(t)dt$$

$$=\displaystyle\int_{0}^{\infin}t(-\dfrac{1}{t^2})dt=\lim\limits_{x\to \infin}[-\ln t]\begin{matrix} x \\ 1 \end{matrix}=-\infin$$

$$E[T^2]=\displaystyle\int_{-\infin}^{\infin}t^2f(t)dt=\displaystyle\int_{0}^{\infin}t^2(-\dfrac{1}{t^2})dt$$

$$=\lim\limits_{x\to \infin}[- t]\begin{matrix} x \\ 1 \end{matrix}=-\infin$$

$$Var(T)=E[T^2]-(E[T])^2=-\infin$$