EX 4.17 breakdown voltage of diode

The breakdown voltage of a randomly chosen diode of a particular type is known to be normally distributed.

Question

What is the probability that a diode’s breakdown voltage is within 1 standard deviation of its mean value?

This question can be answered without knowing either $\mu$ or $\sigma$ , as long as the distribution is known to be normal; the answer is the same for any normal distribution:

$$\begin{array}{rl}& P\left(X\text{ is within 1 standard deviation of its mean}\right)\\ & =P\left(\mu -\sigma \le X\le \mu +\sigma \right)\\ & =P\left(\frac{\mu -\sigma -\mu }{\sigma }\le Z\le \frac{\mu +\sigma -\mu }{\sigma }\right)\\ & =P\left(-1.00\le Z\le 1.00\right)\\ & =\Phi \left(1.00\right)-\Phi \left(-1.00\right)\\ & =0.6826\end{array}$$

The probability that $X$ is

• within 2 standard deviations of its mean is $P\left(-2.00\le Z\le 2.00\right)=.9544$
• within 3 standard deviations of the mean is $P\left(-3.00\le Z\le 3.00\right)=.9974$