The Standard Normal Distribution

Let Z be a normal random variable with mean 0 and variance 1; that is,
                              Z~N(0, 1)
We say that Z follows the standard normal distribution.

Denote the cumulative distribution function as F(z) and a and b as two numbers with a < b; then
                 P(a<Z<b)= F(b) – F(a)

We can obtain probabilities for any normally distributed random variable by first converting the random variable to the standard normally distributed random variable, Z. There is always a direct relationship between any normally distributed random variable and Z. That relationship buses the transformation

This important result allows us to use the standard normal table to compute probabilities associated with any normally distributed random variable. Now let us see how probabilities can be computed for the standard normal Z.

The cumulative distribution function of the standard normal distribution is tabulated in Table 1 in the Appendix. This table gives values of

                       F(z) = P(Z

for non-negative values of z. For example, the cumulative probability for a Z value of 1.25 from Table 1 is

                            F(1.25) = 0.8944

This is the area, designated in the following Figure, for Z less than 1.25. Because of the symmetry of the normal distribution, the probability that Z > -1.25 is also equal to .8944. In general, values of the cumulative distribution function for negative values of Z can be inferred using the symmetry of the probability density function.

To find the cumulative probability for a negative Z (for example, Z = -1.0) defined as

we use the complement of the probability for Z = +1, as shown in the following Figure.

From symmetry we can state that

The following Figure indicate the symmetry for the corresponding positive values of Z.

Figure: Standard normal distribution           Figure: Normal density function
for positive value Z equal 1                          with symmetric upper and lower values