MATH 141-501
Section 8.5 & Section 8.6 Lecture Notes
Continuous Probability Distributions
Recall: We have seen histograms for probability distributions for finite discrete random variables, like the one given below.
As we take a large number of different outcomes and observations, it makes
sense to “smooth out” the distribution by replacing the jagged rectangles with
a smooth curve. This is called a continuous probability distribution.
Example: Approximate the discrete histogram below with a continuous
probability distribution.
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Standard Normal Distribution
One of the most commonly used continuous probability distributions is the
standard normal distribution.
The standard normal distribution looks like this:
Properties of the Standard Normal Distribution
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Normal Distributions
In general, shape of a normal distribution depends on the mean (center) and
standard deviation (spread). Here are some other pictures of normal distributions.
For instance, the distribution below has µ = 100 and σ = 10.
You can “play” with different normal curves at the applet below:
http://www.intmath.com/counting-probability/normal-distribution-graph-interactive.php
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Properties of Normal Distributions
Here is what a normal distribution with mean µ and standard deviation σ
looks like.
Here are some important properties of a normal distribution.
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Example - Weight of Squirrels
The weight, in pounds, of a certain species of adult squirrel is normally
distributed with a mean of 3 pounds and a standard deviation of 0.5 pounds.
1. Let X be continuous probability distribution associated with the weight
of this species of squirrel. Sketch the graph of X.
2. Shade the area on the graph which corresponds to P (X ≤ 3.5).
3. Describe “P (X ≤ 3.5)” in words.
4. What is the probability that a randomly selected squirrel has weight
greater than 3 pounds?
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Useful Calculator Commands for Normal Distribution
Once we set up a problem involving the normal distribution, we can use the
calculator to solve the problem.
The commands for this are summarized in the table below.
If X is a normal probability distribution with mean µ and standard deviation
σ, then
Command
Mathematics
normalcdf(a, b, µ, σ )
P (a ≤ X ≤ b)
invNorm(p, µ, σ )
a such that P (x ≤ a) = p
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Words
Probability that
X takes a value
between a and b
Value for p’th
percentile
Example: The standard normal distribution Z has mean µ = 0 and standard deviation σ = 1. Find the following.
1. P (Z ≤ 0.5)
2. P (Z > 1)
3. P (−1 ≤ Z ≤ 1)
4. P (−1 < Z < 1)
5. The value of a such that P (Z ≤ a) is 0.9.
6. The value of a such that P (Z ≥ a) is 0.9.
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Example: MarbleCo is interested in producing marbles which always have
the same diameter. The company’s statisticians have determined that the marble diameter is normally distributed with a mean of 1.5cm and a standard
deviation of 0.01cm. A marble is considered acceptable if its diameter is in the
range [1.48, 1.52].
1. What percentage of the company’s marbles are acceptable?
2. What is the probability that a marble will have a diameter greater than
1.5?
3. Determine how large a marble’s diameter must be to be in the 99th percentile of marbles.
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Example: Assume that height of American males is normally distributed
with mean 70 inches and standard deviation of 3 inches. Assume that the height
of American females is normally distributed with mean 67 inches and standard
deviation of 3 inches.
• Sketch the continuous distribution function for the height of American
males.
• Find the probability that a randomly selected American male has a height
of less than 65 inches.
• Find the probability that a randomly selected American male is at least
six feet tall.
• Find the probability that a randomly selected American female has a
height of between 62 and 68 inches.
• What height constitutes the 90th percentile for American males? What
height constitutes the 50th percentile for American females?
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