The hardness of a piece of ceramic is proportional to the firing time. Assume that a rating system has been devised to rate the hardness of a ceramic piece and that this measure of hardness is a random variable that is distributed uniformly between 0 and 10. If a hardness in [5,9] is desirable for kitchenware, what is the probability that a piece chosen at random will be suitable for kitchen use?

Solution

Let $X$ denote the rating of the hardness of a ceramic piece.

$X\sim U(0,10)$.

The pdf of $X$ is

$$ \begin{aligned} f(x)&= \frac{1}{10-0}, \;0\leq x \leq 10\\ &= \frac{1}{10} ,\;0\leq x\leq 10 \end{aligned} $$

If a hardness in [5,9] is desirable for kitchenware, the probability that a piece chosen at random will be suitable for kitchen use is

$$ \begin{aligned} P(5< X< 9) &= \int_{5}^{9} f(x) \; dx\\ &= \frac{1}{10}\int_{5}^{9} \; dx\\ &= \frac{1}{10}\big[x\big]_{5}^{9} \\ &= \frac{1}{10}\big[ 9-5]\\ &= 0.4 \end{aligned} $$

Further Reading