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STAT 100A
Chapter 6: The Normal Probability Distribution
Section 6.1: Probability Distributions for Continuous Random Variables
Continuous Probability Distributions
To find probabilities for continuous random
variables, we do not use probability distribution
tables, histograms, or functions (as we did for
discrete random variables). Instead, we use
probability density functions.
The area under the graph of a density function over
some interval represents the probability of observing
a value of the random variable in that interval.
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1. The area under the graph of the pdf over all possible
values of the random variable must equal one.
2. The graph of the pdf must be greater than or equal to
zero for all possible values of the random variable.
That is, the graph of the pdf must lie on or above the
horizontal axis for all possible values of the random
variable.
Uniform Probability Distribution
Requirements for a Continuous
Probability Distribution
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Example: Suppose the reaction time X (in minutes) of a
certain chemical process follows a Uniform Probability
Distribution with 5 ≤ X ≤ 10.
a) Draw the graph of the density curve.
b) What is the probability that the reaction time is between
6 and 8 minutes?
c) What is the probability that the reaction time is less than
6 minutes?
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STAT 100A
Section 6.2: The Normal Probability Distribution
The Normal Distribution
Properties of the Normal Density Curve
1. It is bell-shaped.
2. It is symmetric about its mean.
3. The area under the curve is one.
4. It has inflection points one standard deviation
from the mean in each direction.
5. The Empirical Rule:
1) approx. 68% of the area under the normal
curve is within one standard deviation of the
mean.
2) approx. 95% of the area under the normal
curve is within two standard deviations of
the mean.
3) approx. 99.7% of the area under the normal
curve is within three standard deviations of
the mean.
