Download x - cloudfront.net

Chapter 7
Lesson 7.3
Random Variables and Probability
Distributions
7.3 Probability Distributions for Continuous
Random Variables
Probability Distributions for
Continuous Random Variables
Consider the random variable:
x = the weight (in pounds) of a full-term
newborn child
Suppose that weight is reported to the nearest
pound. The
following
probability
histogram
What
type
of
variable
is
this?
If
weight
is
measured
with
greater
What
is
the
sum
of
the
areas
of all
displays
the
distribution
of
weights.
The
area
of
the
rectangle
and
greater
accuracy,
thecentered
histogram
the
rectangles?
Notice
that
the
rectangles
are
The
shaded
area
represents
the
over
7
pounds
represents
the
This
is
an
example
approaches
ahistogram
smooth
curve.
Nownarrower
suppose
that
and
the
weight
is
reported
begins
to
to
the
probability
6
<
x
<
8.
probability
6.5 appearance.
< xof
< 7.5
a density
curve.
have
a
smoother
nearest 0.1 pound. This would be
the probability
histogram.
Probability Distributions for
Continuous Variables
• Is specified by a curve called a density
curve.
• The probability of observing a value in a
particular interval is the area under the
curve and above the given interval.
• The total area under the density curve
equals one.
Let x denote the amount of gravel sold (in tons)
during a randomly selected week at a particular
sales facility. Suppose that the density curve
has a height f(x) above the value x, where
2(1  x ) 0  x  1
f (x )  
otherwise
0
The density curve is
shown in the figure:
Density
2
1
Tons
1
Gravel problem continued . . .
What is the probability that at most ½ ton of
gravel is sold during a randomly selected week?
P(x < ½) = .75
Density
2
The probability would be the
shaded area under the curve and
above the interval from 0 to 0.5.
1
Tons
1
Gravel problem continued . . .
What is the probability that exactly ½ ton of
gravel is sold during a randomly selected week?
P(x = ½) =
2
0
The
probability
would
be
the
area
Since a line segment has NO area,
Density
the curve that
and above
0.5.
thenunder
the probability
exactly
½
ton is sold equals 0.
1
Tons
1
Gravel problem continued . . .
What is the probability that less than ½ ton of
gravel is sold during a randomly selected week?
P(x < ½) = P(x < ½)
Density
2
1
= .75
Does theHmmm
probability
change
. . . This
is
whether different
the ½ is included
or not?
than discrete
probability distributions
where it does change
the probability whether
a value is included or
Tons
not!
1
Suppose x is a continuous random variable
defined as the amount of time (in minutes) taken
by a clerk to process a certain type of
application form. Suppose x has a probability
distribution with density function:
.5 4  x  6
f (x )  
0 otherwise
The following is the graph of f(x), the density
curve:
Density
0.5
4
5
6
Time (in minutes)
Application Problem Continued . . .
What is the probability that it takes more than
5.5 minutes to process the application form?
P(x > 5.5) = .5(.5) = .25
Find the probability by calculating
the area of the shaded region
(base × height).
Density
0.5
4
5
6
Time (in minutes)
Other Density Curves
Some density curves resemble the one
below. Integral calculus is used to find
the area under the these curves.
Don’t worry – we will use tables (with the
values already calculated). We can also
use calculators or statistical software to
find the area.
Homework
• Pg.320: #7.20-22, 25