Question

The lifetime, in years, of some electronic component is a continuous random variable with the density f(x) =   k x4 for x ≥ 1 0 for x < 1. Find k, the cumulative distribution function, and the probability for the lifetime to exceed 2 years.

Answer 1

I suppose the density is supposed to be

$f(x)=\begin{cases}\frac k{x^4}&\text{for }x\ge1\\0&\text{otherwise}\end{cases}$

(since $kx^4$ diverges as $x\to\infty$, and 0 is conveniently left out of the domain)

For this to be a proper density function, it must be positive (it is) and its integral over the support $[1,\infty)$ must evaluate to 1. We have

$\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=\int_1^\infty\frac k{x^4}\,\mathrm dx=-\frac k{3x^3}\bigg|_1^\infty=\frac k3=1$

which means $k=3$.

The distribution function is obtained by integrating the density:

$F(x)=\displaystyle\int_{-\infty}^xf(t)\,\mathrm dt=\begin{cases}0&\text{for }x$

The probability that the component exceeds 2 years in its lifetime is

$P(X>2)=1-P(X\le 2)=1-F(2)=1-\left(1-\dfrac18\right)=\dfrac18$