Mathematics: 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.

367 meters talldivide 220.2 by 0.6...

The lifetime, in years, of some electronic component

joy3449

23.01.2022

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

$P(X2)=1-P(X\le 2)=1-F(2)=1-\left(1-\dfrac18\right)=\dfrac18$

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