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Math 141 Lecture Notes
Section 8.5 The Normal Distribution
Probability Density Functions
Binomial probability distributions are associated with finite random variables. Normal
probability distributions are associated with continuous random variables.
A probability density function associated with the probability distribution for a
continuous random variable has the following properties:
1. f(x) is nonnegative for all values of x.
2. The area of the region between the graph of f and the x-axis is equal to 1.
Normal Distributions
The graph of a normal distribution, which is bell shaped, is called a normal curve.
The normal curve (and therefore the corresponding normal distribution) is completely
determined by its mean  and standard deviation . The normal curve has the following
characteristics:
1. The curve has a peak at x = .
2. The curve is symmetric with respect to the vertical line x = .
3. The curve always lies above the x-axis but approaches the x-axis as x extends
indefinitely in either direction.
4. The area under the curve is 1.
5. For any normal curve, 68.27% of the area under the curve lies within 1 standard
deviation of the mean (that is, between  -  and  + ), 95.45% of the area lies
within 2 standard deviations of the mean, and 99.73% of the area lies within 3
standard deviations of the mean.
Computations of Probabilities Associated with Normal Distributions
Table 2, Appendix C, gives the areas of the regions under the standard normal curve to
the left of the number z. These areas correspond to probabilities of the form P(Z < z) or
P(Z  z).
Example 1: Let Z be the standard normal variable. By first making a sketch of the
appropriate region under the standard normal curve, find the values of
a. P(Z < 2.14)
b. P(Z > 0.8)
c. P(0.75 < Z < 1.64)
d. P(-1.34 < Z < 2.31)
Example 2: Let Z be the standard normal variable. Find the value of z if z satisfies
a. P(Z < z) = .9115
b. P(Z > z) = .9474
c. P(-z < Z < z) = .6778
The area of the region under the normal curve (with random variable X) between x = a
and x = b is equal to the area of the region under the standard normal curve between z =
(a - )/ and z = (b - )/. In terms of probabilities associated with these distributions, we
have
b
a
P ( a  X  b)  P
Z

 
 
Example 3: Suppose X is a normal random variable with  = 100 and  = 20. Find the
values of
a. P(X < 110)
b. P(X > 80)
c. P(70 < X < 115)