Question

1. Suppose that the error in the reaction temperature, in ◦C, for a controlled laboratory experiment is a continuous random variable X having the probability density function
f(x)=x2/3 for −1 <x<2,
f(x)=0, elsewhere

(a) Verify that f(x) is a density function
(b) Find P(0 <X≤ 1).

Answer 1

$a)\int_{-\infty}^{\infty}f(x)dx=1\\ \int_{-1}^{2}\frac{2}{3}xdx=[\frac{2}{3}\frac{x^2}{2}]_{-1}^2=1.\text{ So } f(x)\text{ is pdf}.$

$b) P\{0<X\leq 1\}=\int_0^1\frac{2}{3}xdx=[\frac{2}{3}\frac{x^2}{2}]_{0}^1=\frac{1}{3}.$