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4.6 The Normal Distribution (or The Gaussian Distribution)
This is the most important continuous distribution. Most real life data follow the
Normal Distribution, for example; height, weight, exam scores etc.
Properties of the Normal Distribution:
1) The pdf of a normal random variable with mean µ and variance σ2is given by
f(x) =
1
2πσ
−
2
e
( x −µ )2
2σ 2
,
−∞ ≤ x ≤ ∞
2) The graph of the distribution is a bell shaped curve centered at µ, and has peak
at x = µ. It is often called the location parameter. Similarly the spread in the
curve is determined by σ, and therefore we call σ the shape parameter. The
larger the value of σ, the more spread out the curve is. If a random variable X
follows the normal distribution with mean µ and variance σ2, we write
X ~ N( µ, σ2).
3) The mean, median and the mode all have the same value for the distribution.
We first look at a special distribution, called the standard normal
distribution. Represented by Z this is the normal distribution with mean 0
and variance 1. What do you think the pdf of Z is?
Probabilities for Z ……..
Examples on the Normal Distribution:
With the standard normal distribution, we now do the opposite of what we
have been doing:
If we are given the probability, we can find the
corresponding point.
a) Find a value, a, such that P( -a < Z < a) = 0.95
b) Find a value b such that Z exceeds it is 0.05.
c) Find a value b, such that Z exceeds it is 0.9.
Suppose now we have a Normal distribution that is not standard normal, i.e.,
X ~ N(µ, σ2). In order to find probabilities for X, we can standardize it.
If X ~ N(µ, σ2), then
Z=
(X − µ)
~ N(0,1)
σ
and so
P(X < c) = P(
X−µ c−µ
c−µ
<
) = P( Z <
).
σ
σ
σ
examples : Suppose X ~ N(10, 4).
Find P(7<X<13) =
= P( _______ < Z < ___________ )
=
= 0.8664
1) Studies show that gasoline use for compact cars is normally
distributed with mean µ = 25.5 mpg and standard deviation, σ = 4.5 mpg.
a) Find the percentage of cars that have mileage more than
30 mpg?
b) Suppose that a manufacturer wants to produce a compact
car that outperforms 99% of cars. What should the gasoline mileage of this
car be?
X0 = 35.985 mpg.
2) Scores on the pharmacy board exam are normally distributed with
a mean of 75 and a standard deviation of 23.
a) P( 65 < X < 85) = P( ________ < Z < _________ ) = 0.34
b) Find the 95 th percentile of the scores -
......... a = 113.