Math 141: Business Mathematics I
Fall 2015
§8.5-6 The Normal Distribution
Instructor: Yeong-Chyuan Chung
In the previous sections, we have been dealing with finite discrete random variables. Recall
(from §8.1) that these are random variables that can take on only finitely many values. In
this section, we will consider continuous random variables of a particular type. These are
random variables that can take on infinitely many values within an interval.
For finite discrete random variables, we can tabulate all the possible values and associated
probabilities, thereby getting the probability distribution. We can also represent the distribution using a histogram. For continuous random variables, it is impossible to list all
the possible values because there are infinitely many, but we can represent the probability
distribution using a probability density function.
A probability density function f has the following properties:
1. It is defined on the interval of values taken on by the random variable, and it takes
nonnegative values.
2. The area of the region between the graph of f and the x-axis is equal to 1.
3. The probability that the random variable takes a value in an interval between a and
b, denoted by P (a < X < b), is given by the area of the region between the graph of
f and the x-axis from x = a to x = b.
For any continuous random variable X and any value a, the probability P (X = a) is 0
because the corresponding area is 0. As a consequence, we have
P (a < X < b) = P (a < X ≤ b) = P (a ≤ X < b) = P (a ≤ X ≤ b).
In other words, for continuous random variables, it does not matter whether the endpoints
of the interval under consideration are included.
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§8.5-6 The Normal Distribution
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The mean and standard deviation of a continuous random variable have essentially the
same meanings as the mean and standard deviation of a finite discrete random variable.
Now we will consider a particular type of continuous distribution. These distributions are
completely determined by their means and standard deviations, and they are the normal
distributions.
The graph of a normal distribution is bell shaped, and is sometimes called a normal curve.
The curve is completely determined by the mean µ and the standard deviation σ. More
precisely, the curve has the following characteristics:
1. The area under the curve is 1. (This is true for all continuous random variables.)
2. The curve always lies above the x-axis. (This is also true for all continuous random
variables.)
3. The curve peaks at x = µ.
4. The curve is symmetric about the vertical line x = µ.
The normal curve with mean µ = 0 and standard deviation σ = 1 is called the standard
normal curve. The corresponding distribution is called the standard normal distribution, and the associated random variable is called the standard normal random variable,
usually denoted by Z.
Using the graphing calculator in problems involving normal distributions
There are two commands in the calculator that can be used in problems involving normal
distributions - normalcdf and invnorm. Both can be found by pressing 2nd VARS.
normalcdf is used to compute probabilities involving a normal random variable X.
The syntax is normalcdf(a,b,µ,σ) where a is the lower bound, b is the upper bound, µ is
the mean, and σ is the standard deviation. If the interval under consideration has no lower
bound, then we type -1E99 in place of a. Note that “E” is typed by pressing (2nd ,). If the
interval has no upper bound, then we type 1E99 in place of b.
§8.5-6 The Normal Distribution
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invnorm is used to find an upper bound for the standard normal random variable corresponding to a given probability (or a value a such that the area of the graph to the left of
the line x = a is the given one). In other words, it is used to solve problems of the form
“Given that P (Z < z) = . . ., find the value of z.”
The syntax is invnorm(area,µ,σ) where µ is the mean, and σ is the standard deviation.
Example (Exercise 7,8,14 in the text). For each of the following, sketch the area under the
standard normal curve corresponding to the given probability, and compute the probability.
(a) P (Z < 1.38)
(b) P (Z > 2.27)
(c) P (−1.41 < Z < −0.24)
Example (Exercise 18 in the text). Suppose X is a normal random variable with µ = 380
and σ = 20. Find the value of
(a) P (X < 405)
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(b) P (400 ≤ X < 430)
(c) P (X > 400)
Example (Exercise 16 in the text). Let Z be the standard normal random variable. Find
the values of z if z satisfies:
(a) P (Z < z) = .36
(b) P (Z > z) = .9678
(c) P (−z < Z < z) = .8354
§8.5-6 The Normal Distribution
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Although we will not discuss this in detail, it is good to know that under certain conditions,
we can approximate a binomial distribution by a normal distribution. Just imagine smoothing out the histogram we get from a binomial distribution to get a smooth curve. In the
past, before graphing calculators became popular, this approximation technique would have
been a way to simplify computations involving binomial distributions.
The normal distribution itself is a good model in many real-life situations. We typically
expect quantities like heights and weights of a certain population to be normally distributed.
The scores on a test can often be modeled quite well by a normal distribution.
Example (Exercise 10 in the text). The scores on an economics examination are normally
distributed with a mean of 72 and a standard deviation of 16. If the instructor assigns a
grade of A to 10% of the class, what is the lowest score a student may have and still obtain
an A?
Example (Exercise 2 in the text). According to the data released by the Chamber of Commerce of a certain city, the weekly wages of factory workers are normally distributed with a
mean of $720 and a standard deviation of $60. What is the probability that a factory worker
selected at random from the city makes a weekly wage:
1. Of less than $720?
2. Of more than $912?
3. Between $660 and $780?