A6 Some Important Theoretical Probability Distributions

In the text extensive use is made of the following probability distributions.

Normal Distribution

The best known of all the theoretical probability distributions is the normal distribution, whose bell-shaped picture is familiar to anyone with a modicum of statistical knowledge.


A (continuous) random variable X is said to be normally distributed if its PDF has the following form:

f (x) = —— exp — --2— —to < x < to a\/2n V 2 a2 /

where m and a2, known as the parameters of the distribution, are, respectively, the mean and the variance of the distribution. The properties of this distribution are as follows:

1. It is symmetrical around its mean value.

2. Approximately 68 percent of the area under the normal curve lies between the values of m ± a, about 95 percent of the area lies between M ± 2a, and about 99.7 percent of the area lies between m ± 3a, as shown in Figure A.4.

3. The normal distribution depends on the two parameters m and a2, so once these are specified, one can find the probability that X will lie within a certain interval by using the PDF of the normal distribution. But this task can be lightened considerably by referring to Table D.1 of Appendix D. To use this table, we convert the given normally distributed variable X with mean m and a2 into a standardized normal variable Z by the following transformation:

An important property of any standardized variable is that its mean value is zero and its variance is unity. Thus Z has zero mean and unit variance. Substituting z into the normal PDF given previously, we obtain f (Z) =ybexp (—2 Z2

FIGURE A.4 Areas under the normal curve.


which is the PDF of the standardized normal variable. The probabilities given in Appendix D, Table D.1, are based on this standardized normal variable.

By convention, we denote a normally distributed variable as

where — means "distributed as," N stands for the normal distribution, and the quantities in the parentheses are the two parameters of the normal distribution, namely, the mean and the variance. Following this convention,

means X is a normally distributed variable with zero mean and unit variance. In other words, it is a standardized normal variable Z.

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