Exercise Solution 3.31

Values of the CDF Φ(z) = Pr(Z ≤ z) correspond to areas under the PDF. These can be looked up in a standard normal table. The table only lists values Φ(z) for positive z, but consider the following exhibit.

Exhibit s1: The desired probability Φ(–.51) = Pr(X ≤ –.51) equals the area of the indicated shaded region in the left chart. By symmetry, that area equals the area indicated in the right chart, which equals 1 – Φ(.51). A standard normal table provides a value for Φ(.51).

By the symmetry of the normal distribution and the fact that the area under the entire PDF is 1, we obtain

[s1]

[s2]

[s3]

[s4]

Exhibit s2: The value Φ(0.51) = 0.6950 can be obtained from a standard normal table.