4.3.1 The Standard Normal Distribution

The computation of $P\left(a\le X\le b\right)$ when $X$ is a normal rv with parameters $\mu$ and $\sigma$ requires evaluating

$${\int }_a^b\frac{1}{\sqrt{2\pi }\sigma }{e}^{-{\left(x-\mu \right)}^2/\left(2{\sigma }^2\right)}dx\text{\hspace{1em}(4.4)}$$

None of the standard integration techniques can be used to accomplish this. Instead, for $\mu =0$ and $\sigma =1$ , Expression (4.4) has been calculated using numerical techniques and tabulated for certain values of $a$ and $b$. This table can also be used to compute probabilities for any other values of $\mu$ and $\sigma$ under consideration.

Definition

The normal distribution with parameter values $\mu =0$ and $\sigma =1$ is called the standard normal distribution. A random variable having a standard normal distribution is called a standard normal random variable and will be denoted by $Z$. The pdf of $Z$ is

$$f\left(z;0,1\right)=\frac{1}{\sqrt{2\pi }}{e}^{-z^2/2}-\infty <z<\infty$$

The graph of $f\left(z;0,1\right)$ is called the standard normal (or $z$ ) curve. Its inflection points are at 1 and -1. The cdf of $Z$ is $P\left(Z\le z\right)={\int }_{-\infty }^{z}f\left(y;0,1\right)dy,$ which we will denote by $\Phi \left(z\right)$ .

The standard normal distribution almost never serves as a model for a naturally arising population. Instead, it is a reference distribution from which information about other normal distributions can be obtained. Appendix Table A.3 gives $\Phi \left(z\right)=P\left(Z\le z\right)$, the area under the standard normal density curve to the left of $z$ , for $z=-3.49$, $-3.48$, $\dots$, $3.48$, $3.49$. Figure 4.14 illustrates the type of cumulative area (probability) tabulated in Table A.3. From this table, various other probabilities involving $Z$ can be calculated.

Figure 4.14 Standard normal cumulative areas tabulated in Appendix Table A.3

EX 4.13