Download Algebra I - Fort Thomas Independent Schools

AP Statistics
Notes
Name: ____________
Date: _____________
Lesson 7.1B: Continuous Random Variables
Objectives:
B:
Recognize and define a continuous random variable, and determine probabilities of
C:
events as areas under density curves.
Given a Normal random variable, use the standard Normal table or a graphing
calculator to find probabilities of events as areas under the standard Normal
distribution curve.
Vocabulary:
Continuous random variable
1.
Definition of a Continuous Variable
Ex#1: In an engineering stress test, pressure is applied to a thin 1-foot-long bar until the
bar snaps. There is uncertainty concerning the precise location at which the bar
will snap. Let x be the distance from the left end of the bar to the break.
Then x = .25 is one possibility, and x = .90 is another.
Name two other possibilities. ________ and ________
In fact, any number between ___ and ___
is a possible value of x.
The variable, x, is a continuous random variable because ___________________
___________________________________________________________________.
Ex#2: Define a continuous random variable, X, by the following:
X = the amount of time (min.) taken by a clerk to process
a certain type of application form
The probability distribution of a continuous random variable X is described
by a density curve.
Suppose that X has a probability distribution with density function
𝑓(𝑥) = {
0.5 𝑓𝑜𝑟 4 < 𝑋 < 6
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Draw the probability distribution of X.
Note: The probability distribution for
this problem is called a
uniform distribution.
The probability of an event is found by __________________________________
___________________________________________________________________.
Find P(4.5 < X < 5.5) __________
Find P(4.5  X  5.5) __________
Find P(X > 5.5) __________
Find P(4.8  X) __________
T/F
2.
The probability of a single outcome is always 0.
Normal Distributions
We have already spent some time studying the density function of one particular type of
continuous random variable, namely, a normal random variable. To review, consider
the following example.
Ex#3: A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose
that the variable x = actual capacity of a randomly selected tank has a distribution
that is well approximated by a normal curve with mean 15.0 gallons and standard
deviation 0.1 gallon.
a.
What is the probability that a randomly
selected tank will hold at most 14.8 gal?
b.
What is the probability that a randomly
selected tank will hold at least 15.25 gal?
c.
What is the probability that a randomly
selected tank will hold between
14.7 and 15.1 gal?
d.
If two such tanks are independently selected,
what is the probability that both hold at most 15 gal?