Download 7.1 - Continuous Probability Distribution and The Normal Distribution

7.1 - Continuous Probability Distribution and The Normal Distribution
Since a continuous random variable x can assume an infinite number of uncountable values, we have
to look at x assuming a value within an interval.
The probability distribution of a continuous random variable is often presented in the form of a probability
density curve (also called probability distribution curve). The total area under the curve must equal 1.
The area under the curve between two values a and b has two interpretations:
1. It is the proportion of the population whose values are between a and b.
2. It is the probability that a randomly selected individual will have a value between a and b.
The probability that a continuous random variable x assumes a value within a certain interval is given
by the area under the curve between the two limits of the interval.
The probability that a continuous random variable x assumes a single value is always zero. It follows
from this that P (a ≤ x ≤ b) = P( a < x < b) .
A continuous random variable can have many different distributions. One of the most widely used
distributions is the normal probability distribution.
A Normal Probability Distribution gives a bell-shaped curve, symmetric about the mean, and with
the two tails of the curve extending indefinitely. The total area under the curve is 1.0.
2
1
e − (1/ 2)[( x − µ )/σ ]
σ 2π
Luckily, we will never have to use this equation in our class.
The equation of the normal distribution curve is
f ( x) =
The parameters of the normal distribution is _______ and ________ .
There is not just one normal distribution curve but a whole family of them, each one depending on µ
and σ . A larger standard deviation gives a wider curve than a smaller standard deviation, while the
mean decides where to center the curve.
The standard normal distribution is a normal distribution with µ = 0 and σ = 1 .
The random variable that has a standard normal distribution is denoted z. The units marked on the
horizontal axis are denoted z and are called the z-values.
A z-value gives the distance from the mean in terms of number of standard deviations. That is, z = 2
is located 2 standard deviations to the right of the mean (right because it is a positive number), and
z = -1.5 is located 1.5 standard deviations to the left of the mean (left because it is a negative
number).
We can look up probabilities for different z-values using a table or by using our calculator. In this
class we will use our calculators. This is how to do it on a TI-83/84:
Press PRGM -> NORMAL83 ->ENTER -> ENTER -> choose the option that fits your specific
problem and enter mean = 0, standard deviation = 1, as well as your limit(s) -> ENTER
We can also reverse the procedure and find the z-value(s) corresponding to a known area under the
standard normal distribution curve rather than vice versa.
We can find z-values using our TI-83/84 calculator:
Press PRGM -> INVNOR83 ->ENTER -> ENTER -> choose the option that fits your specific
problem and enter the known area.
We Know
Calculator Program
We Get
z-value
NORMAL83
Area / Probability
Area / Probability
INVNOR83
z-value
For the chapter 7 homework, as well as on any quizzes and exams, use the appropriate calculator
program rather than the tables in the appendices to calculate any probabilities, but be sure to include
what program you are using, the parameters, and an appropriate, labeled picture.
Find the following:
(a) the area under the normal curve
from z = .23 to z = 2.10
(f) P ( z ≥ 7.65)
(b) the area under the normal curve
to the left of z = -.45
(g) P ( z = −1.38)
(c) P ( −2 ≤ z ≤ 2)
(h) the z-score for which the area to
its right is 0.25.
(d) P ( −0.40 ≤ z )
(i) the z-scores that bound the
middle 70% of the area under the
normal curve.
(e) P ( z ≤ −0.40)