CHAPTER 5
CONTINUOUS PROBABILITY DISTRIBUTIONS
1. Continuous Probability Functions (Probability Density
Functions)
In the last chapter, we discussed discrete random variables and their probability
distribution functions. We were able to define these functions using tables because
we were able to list out all of the possible data values. However, we can’t list all
the possible outcomes of a continuous random variable, because the range of data
values are all the values in an interval on the number line.
The probability distribution of a continuous random variable, X, is defined by
its probability density function (pdf ) or density curve, f (x). The most
important fact about density curves is:
The area under the density curve, f (x), corresponds to probabilities.
Example 1. Probability = area under density curve. Draw the “perfect
bell-shaped distribution” using a smooth line (rather than histogram bars) and with
the mean, µ = 4 in the center of the curve on the x−axis. This is what the density
curve for the normal distribution looks like.
(1) Shade the area that equals P (3 < x < 5)
(2) Shade the area that equals P (x < 2.5)
(3) Shade the area that equals P (x > 6)
note: For continuous distributions, P (3 < x < 5) = P (3 ≤ x ≤ 5). In fact,
P (x = 3) = 0 because the area under the curve at a single number has no width.
also note: Calculus is required to find areas under the normal density curve. We
will use technology (calculators) to find these areas.
A function f (x) must have the following properties to be considered a “valid”
probability density curve:
(1) The entire curve must be above the x−axis.
.
(2) The TOTAL area under the curve must be
A few more facts about density curves:
• The x−axis represents the range of data values.
• The y−axis has decimal values, like relative frequencies.
• The curve is smooth, but otherwise matches a “perfect histogram” of the
data.
• We will find probabilities of intervals of numbers, rather than exact numbers:
– P (3 < x < 5) = the probability that a randomly selected data value
is between 3 and 5
– P (x < 2.5) = the probability that a randomly selected data value is
at most 2.5
1
2
CONTINUOUS PROBABILITY DISTRIBUTIONS
– P (x > 6) = the probability that a randomly selected data value is
more than 6
2. The Uniform Distribution
The uniform distibution is the simplest continuous probability distribution.
X is a uniform random variable if the possible data values range over an
interval of the number line, and every number in that interval is equally likely (not
just whole numbers, but decimals too).
Notation: X ∼ U (a, b) means that random variable X has a uniform distribution
over the interval [a, b].
Example 2. Recognizing a uniform distribution. The data in the following
table are 55 smiling times, in seconds, of eight-week-old babies.
10.4 19.9 18.8 13.9 17.8 16.8 21.6 17.9 12.5
12.8 14.8 22.8 20.0 15.9 23.3 13.4 17.1 14.5
1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9
5.8 6.9 23.6 5.8 21.7 11.8 3.4 2.1 4.5
8.9 9.4 9.4 7.6 10.0 3.3 6.7 7.8 11.6
11.1 4.9
19.0 22.8
19.2 9.8
6.3 10.7
13.8 18.6
Complete the frequency/relative frequency table for the smiling times data.
x
0-4
4-8
8-12
12-16
16-20
20-24
frequency rel freq
6
0.109
11
0.2
13
0.236
8
10
(If a data value falls on a boundary value, put it into the lower class.)
(1) Use words to describe random variable X: X =
(2) Is X continuous or discrete?
(3) Does X appear to have a uniform distribution? In other words:
(a) What appears to be the range of possible data values, x?
(b) Do all of the data values in this range appear to be equally likely?
(4) X ∼
Example 5.2
Since all the data values are equally likely, the mean of X ∼ U (a, b) is simply
the midpoint of the range of possible data values:
a+b
.
2
The standard deviation of X ∼ U (a, b) less intuitive, but still a simple calculation:
r
(b − a)2
σ=
12
µ=
Example 3. Mean of the uniform distribution. Suppose X ∼ U (0, 24), the
length of smiles (in seconds) of eight-week-old babies (from Example 2).
3
(1) Find the theoretical mean smiling time of this eight-week-old baby. Use
the correct notation.
(2) Use the frequency table from Example 2 to estimate the mean of the sample
data. Use the correct notation.
(3) Are the theoretical and sample means the same? Why or why not?
Example 5.2 continued.
Example 4. Height of a uniform density curve. Suppose X ∼ U (0, 24), the
length of smiles (in seconds) of eight-week-old babies (from Example 2).
(1) Recall that a density “curve” is a smooth line that follows the shape of a
“perfect histogram”. Sketch the density curve for X. Make sure the curve
starts and stops at the correct x values, but don’t worry about the height
of the curve for now.
.
(2) The area under the curve must be
(3) The area under the curve has the shape of a
.
Use what you know about this shape to find the height of the curve and
label it on the y−axis.
(4) Use what you learned from this example to make a general formula for
the height of the probability density curve for uniform random variable
X ∼ U (a, b).
Example 5.2 continued.
If X ∼ U (a, b), the probability density curve f (x) is a
from
to
on the x−axis at a height of
. In mathematical notation, we write
1
f (x) =
, where a ≤ x ≤ b.
b−a
(You will use this notation in the homework.)
For continuous distributions, probabilities are areas under the density curve. In
particular, uniform distribution probabilities are areas of rectangles, which we
know all about.
Example 5. Uniform distribution probabilities. Suppose X ∼ U (0, 24), the
length of smiles (in seconds) of eight-week-old babies (from Example 2).
(1) What is the probability that a randomly chosen eight-week-old baby smiles
between 2 and 18 seconds.
(2) Find the 90th percentile for an eight-week-old baby’s smiling time.
(3) Find the probability that a random eight-week-old baby smiles more than
12 seconds knowing that the baby smiles more than eight seconds. Hint:
Consider another uniform distribution over the restricted sample space of
8 to 24 seconds.
(Use the correct notation for probabilities.)
Example 5.3.
Example 6. Suppose the time it takes a nine-year-old child to eat a donut is
between 0.5 and 4 minutes, inclusive. Let X = the time, in minutes, it takes
a nine-year-old child to eat a donut, and that X is uniformly distributed: X ∼
.
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CONTINUOUS PROBABILITY DISTRIBUTIONS
(1) Find the mean amount of time it takes a nine-year-old child to eat a donut.
Use the correct notation for the mean.
(2) The probability that a randomly selected nine-year-old child eats a donut
in at least two minutes is
.
(3) Find the probability that a different nine-year-old child eats a donut in
more than two minutes, given that the child has already been eating the
donut for more than 1.5 minutes.
(Use the correct notation for probabilities.)
Example 5.5