3.4. The Normal Distribution 191normtab <- function(){ za <- seq(0.00,3.59,.01);
pz <- t(matrix(round(pnorm(za),digits=4),nrow=10))
colnames(pz) <- seq(0,.09,.01)
rownames(pz) <- seq(0.0,3.5,.1); return(pz)}The functionnormtabcan be downloaded at the site mentioned in the Preface.
Example 3.4.2(Empirical Rule).LetXbeN(μ, σ^2 ). Then, by Table II or R,
P(μ− 2 σ<X<μ+2σ)=Φ(
μ+2σ−μ
σ)
−Φ(
μ− 2 σ−μ
σ)=Φ(2)−Φ(−2)
=0. 977 −(1− 0 .977) = 0. 954.Similarly,P(μ−σ<X<μ+σ)=0.6827 andP(μ− 3 σ<X<μ+3σ)=0.9973.
Sometimes these three intervals and their corresponding probabilities are referred
to as theempirical rule. Note that we can use Chebyshev’s Theorem (Theorem
1.10.3), to obtain lower bounds for these probabilities. While the empirical rule is
much more precise, it also requires the assumption of a normal distribution. On the
other hand, Chebyshev’s theorem requires only the assumption of a finite variance.
Example 3.4.3.Suppose that 10% of the probability for a certain distribution that
isN(μ, σ^2 ) is below 60 and that 5% is above 90. What are the values ofμandσ?
We are given that the random variableXisN(μ, σ^2 )andthatP(X≤60) = 0. 10
andP(X≤90) = 0.95. Thus Φ[(60−μ)/σ]=0.10 and Φ[(90−μ)/σ]=0.95. From
TableIIwehave
60 −μ
σ
=− 1. 28 ,90 −μ
σ=1. 64.These conditions require thatμ=73.1andσ=10.2 approximately.
Remark 3.4.1.In this chapter we have illustrated three types ofparameters
associated with distributions. The meanμofN(μ, σ^2 ) is called alocation param-
eterbecause changing its value simply changes the location of the middle of the
normal pdf; that is, the graph of the pdf looks exactly the same except for a shift
in location. The standard deviationσofN(μ, σ^2 ) is called ascale parameter
because changing its value changes the spread of the distribution. That is, a small
value ofσrequires the graph of the normal pdf to be tall and narrow, while a large
value ofσrequires it to spread out and not be so tall. No matter what the values
ofμandσ, however, the graph of the normal pdf is that familiar “bell shape.” In-
cidentally, theβof the gamma distribution is also a scale parameter. On the other
hand, theαof the gamma distribution is called ashape parameter, as changing
its value modifies the shape of the graph of the pdf, as can be seen by referring to
Figure 3.3.1. The parameterspandμof the binomial and Poisson distributions,
respectively, are also shape parameters.
Continuing with the first part of Remark 3.4.1, ifX isN(μ, σ^2 )thenwesay
thatXfollows thelocation modelwhich we write as
X=μ+e, (3.4.13)