§7.1
Properties of the Normal
Distribution
Example 1: Imagine that a friend of yours is always late. In fact, your friend is always a
minimum of 35 to 55 minutes late. Let the random variable X represent the number of
minutes that your friend will be late where x is 1-minute intervals, all of which are
equally likely and x is between x = 35 and x = 55. That is to say your friend is just as
likely to be 41 to 42 minutes late or 52 to 53 minutes late. Because any two intervals of
equal length between 35 and 55, inclusive, are equally likely, the random variable X is
said to follow a uniform probability distribution.
For discrete random variables, we usually substitute the value of the random variable
into a formula to compute probabilities. Things are not as easy for continuous variables
however. Since there are an infinite number of possible outcomes, the probability of
observing a particular value of a continuous random variable is 0. For instance, the
probability that your friend is exactly 38.99735797456 minutes late is 0. Calculating this
probability, there is 1 way for your friend to be exactly this late, out of an infinite
number of possible values between 35 and 55. To deal with this problem, we will
compute probabilities of continuous random variables over intervals.
Definition: A probability density function (pdf) is an equation used to compute
probabilities of continuous random variables. It must satisfy the following two properties:
Two Properties of a Probability Density Function (pdf)
1. The total area under the graph of the equation over all possible values of the random
variable must equal 1.
2. The height of the graph of the equation must be greater than or equal to 0 for all
values of the random variable. That is, the graph of the equation must lie on or
above the horizontal axis for all values of the random variable.
SECTION 7.1
1
Density
Example 1 (continued): Below is the graph of the probability density function for the
random variable X where X = how late your friend is.
0
35
55
X
Time (mins)
OBJECTIVE 1: Utilize the Uniform Probability Distribution
To find the probability of a certain interval, you must find the area of the graph that is
above the interval.
Example 1 (continued): What is the probability that your friend will be between 40 and
45 minutes late?
SECTION 7.1
2
Practice: The reaction time x (in minutes) of a certain chemical process follows a
uniform probability distribution with
(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 8 minutes?
SECTION 7.1
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OBJECTIVE 2: Graph of a Normal Curve
Definition: A continuous random variable is normally distributed, or has a normal
probability distribution, if the relative frequency histogram of the random variable has the
shape of a normal curve.
Each curve is determined by two parameters – its mean, and its standard deviation,
The figures below are different normal curves and show how changing the mean and
standard deviation affect the curve.
In figure (a), there are two normal
density curves. The density curve to the
left has
and the density
curve to the right has
. We
can see that increasing the mean from 0
to 3 caused the graph to shift three units
to the right, but the graphs maintained
their shape since the standard deviations
are the same.
SECTION 7.1
In figure (b), two normal density curves
are drawn. The taller density curve has
and the other has
.
We can see that increasing the standard
deviation from 1 to 2 causes the graph to
become flatter and more spread out, but
maintained its location of center since
their means are the same.
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Characteristics of the Normal Density Curve
1. The mean = mode = median and are located at the center of the distribution, which
is the highest point on the curve.
2. The points at
and
are the inflection points on the normal curve.
These are the points where the curvature changes.
3. The standard deviation of the variable determines the spread of the curve about the
mean.
4. The area under the curve is 1.
5. The curve is symmetric about the mean. The area to the left of the mean is 0.5 and
the area to the right of the mean is 0.5.
6. As x increases, without bound (gets larger and larger), the graph approaches, but
never reaches the horizontal axis. As x decreases without bound (gets smaller and
smaller), the graph approaches, but never reaches the horizontal axis.
7. The Empirical Rule: Approximately 68% of the area under the normal curve is
between
and
. Approximately 95% of the area under the normal
curve is between
and
. Approximately 99.7% of the area
under the normal curve is between
and
.
SECTION 7.1
5
Example 2: Suppose
and
. Graph the normal curve.
OBJECTIVE 3: Standardizing the Normal Curve
We saw that when we change the mean and/or the standard deviation, it changes the
graph of the normal curve. Since the mean and the standard deviation can be any
value, this means that there are an infinite number of different normal curves that we
could potentially have to work with. Instead of this, we will “standardize” each normal
curve to the standard normal curve. In doing this, we will only have to work with one
normal curve, the standard normal curve. The way that we will be standardizing the
normal curves will be through using z-scores.
Standardizing a Normal Random Variable
Suppose the random variable X is normally distributed with mean
Then, the random variable
is normally distributed with mean
the standard normal distribution.
SECTION 7.1
and
and standard deviation
The random variable Z is said to have
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Example 3: A random variable X is normally distributed with
(a) Compute
for
(b) Compute
(c) The area under the first normal curve between
is the area between and
SECTION 7.1
and
and
for
is 0.2880. What
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Example 4: The lives of refrigerators are normally distributed with mean
and standard deviation
years.
years
(a) Draw a normal curve with the parameters labeled. Then, shade the region that
represents the proportion of refrigerators that last for more than 17 years.
(b) Suppose that the area under the normal curve to the right of x = 17 is 0.1151.
Provide two interpretations of this result.
SECTION 7.1
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