Normal Distribution - Section 4.5
A normal distribution is used to model continuous data when the probability histogram has an
approximate bell-shape. The normal distribution has the following properties:
•The parameters for the normal distribution are the mean µand the variance σ2. The standard
normal distribution has mean µ= 0 and variance σ2= 1.
•The probability density function for the normal distribution is
f(y) = 1
σ√2πe−
(y−µ)2
2σ2− ∞ < y < ∞
•The distribution function F(y) for the normal distribution does not have a closed form so-
lution. You must use tables or a computer package to find probabilities associated with the
normal distribution.
Here are graphs of the probability density function and the distribution function of a Normal
distribution with µ= 0 and σ2= 1.
p.d.f. of Normal distribution with mean = 0 and variance = 1
y
f(y)
-4 -2 0 2 4
0.0 0.1 0.2 0.3 0.4
Distribution of Normal distribution with mean = 0 and variance = 1
y
F(y)
-4 -2 0 2 4
0.0 0.2 0.4 0.6 0.8 1.0
•The theoretical mean of the normal distribution is
µ=E(Y) = µ
•The variance of the normal distribution is
σ2=V(Y) = σ2
•The normal distribution is very important in statistical theory and we will be learning much
more about this distribution in Statistics 342.
Working with normal random variables in R.
To find the probability P(Y≤y) the command in R is
pnorm(y,mu,sigma)
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