Download Lecture 9 Uniform and normal distributions

4. Random Variables
A random variable is a way of recording a
quantitative variable of a random experiment.
4. Random Variables
A random variable is a way of recording a
quantitative variable of a random experiment.
In particular Chapter 4 talks about discrete
random variables.
4. Random Variables
A random variable is a way of recording a
quantitative variable of a random experiment.
In particular Chapter 4 talks about discrete
random variables.
If a random variable has a particular
distribution (such as a binomial distribution)
then our work becomes easier. We use
formulas and tables.
5. Continuous Random Variables
A random variable is a way of recording a
quantitative variable of a random experiment.
In particular Chapter 4 talks about discrete
random variables.
If a random variable has a particular
distribution (such as a binomial distribution)
then our work becomes easier. We use
formulas and tables.
5. Continuous Random Variables
A random variable where X can take on a
range of values, not just particular ones.
Examples:
Heights
Distance a golfer hits the ball with their driver
Time to run 100 meters
Electricity usage of a home.
Continuous probability
distribution functions
For a discrete random variable,
probabilities are given as a table of values,
and the distribution can be graphed as a
bar graph.
For a continuous random variable,
probabilities are specified by a continuous
function. The graph of the probability
distribution function is a curve.
Figure 5.1 A probability f(x) for a
continuous random variable x
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Definition
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Figure 5.2 Density Function for Friction
Coefficient, Example 5.1
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Find probability friction is less than 10.
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Find probability friction is less than 10.
Solution:Probabiity = area of shaded
triangle = (1/2)(5)(0.2)=0.5
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Uniform Distribution
A Uniform Distribution has equally likely
values over the range of possible
outcomes.
Uniform Distribution
A Uniform Distribution has equally likely
values over the range of possible
outcomes.
A graph of the uniform probability
distribution is a rectangle with area equal
to 1.
Example
The figure below depicts the probability distribution for
temperatures in a manufacturing process. The
temperatures are controlled so that they range
between 0 and 5 degrees Celsius, and every possible
temperature is equally likely.
P(x)
0.2
0
0
1
2
3
4
Temperature (degrees Celsius)
5
x
Example
Note that the total area under the
“curve” is 1.
P(x)
0.2
0
0
1
2
3
4
Temperature (degrees Celsius)
5
x
Example
P(x)
0.2
0
0
1
2
3
4
5
x
Temperature (degrees Celsius)
What is the Probability that the temperature is
exactly 4 degrees?
Example
P(x)
0.2
0
0
1
2
3
4
5
x
Temperature (degrees Celsius)
What is the Probability that the temperature is
exactly 4 degrees?
Answer: 0
Explanation
Since we have a continuous random variable
there are an infinite number of possible
outcomes between 0 and 5, the probability of
one number out of an infinite set of numbers is
0.
Example
What is the probability the temperature is
between 10C and 40C?
P(x)
0.2
0
0
1
2
3
4
Temperature (degrees Celsius)
5
x
Example
What is the probability the temperature is
between 10C and 40C?
P(x)
0.2
0
0
1
2
3
4
Temperature (degrees Celsius)
5
x
What is the probability the temperature is
between 10C and 40C?
P(x)
0.2
0
0
1
2
3
4
5
x
Temperature (degrees Celsius)
We know that the total area of the rectangle
is 1, and we can see that the part of the
rectangle between 1 and 4 is 3/5 of the total,
so P(1  x  4) = 3/5*(1) = 0.6.
Review: Probabilities and Area
For a density curve depicting the
probability distribution of a continuous
random variable,
– the total area under the curve is 1,
– there is a direct correspondence between
area and probability.
– Only the probability of an event occurring
in some interval can be evaluated.
– The probability that a continuous random
variable takes on any particular value is
zero.
General Uniform Distribution
A Uniform Distribution has equally likely
values over the range of possible
outcomes, say c to d.
1
Height of the density function : f(x) 
d c
cd
Mean   
2
d c
Standard Deviation   
12
Normal Distributions
This is the most common observed
distribution of continuous random variables.
A normal distribution corresponds to bellshaped curves.
Normal Distributions
This is the most common observed
distribution of continuous random variables.
A normal distribution corresponds to bellshaped curves.
y 
e
( x   ) 2

/ 2 2
2
Reminder: Mu is the mean, sigma is the standard deviation.
Examples
The following are examples of normally
distributed everyday data.
– Grades on a test.
– How many chips are in a small bag of potatoe
chips.
– The measurements of distance between two
points.
– The heights of students in this class.
Normal Distributions
Normal Distributions
Shape of this curve is determined by µ and σ
– µ it’s centered, σ is how far it’s spread out.
Standard Normal Distribution
The Standard Normal Distribution is a
normal probability distribution that has a
mean of 0 and a standard deviation of 1.
  0,
1
In this way the formula giving the heights of
the normal curve is simplified greatly.
Z-score
Standard Normal Probabilities
P(0  z  1) represents the probability that
z takes on values between 0 and 1, which
is represented by the area under the
curve between 0 and 1.
P(0  z  1) = 0.3413
P(0  z  1) = 0.341
Revelation!
Since the mean is 0 and the standard
deviation is 1, this tells us that the
probability that z is within one standard
deviation of the mean (either below or
above) is (2)(0.341)= 0.682.
P(0  z  1) = 0.341
Revelation!
Since the mean is 0 and the standard
deviation is 1, this tells us that the
probabiity that z is within one standard
deviation of the mean (either below or
above) is (2)(0.341)= 0.682.
Agrees with Empirical Rule: 68% of
data lies within one standard deviation
of the mean
Finding Probabilities when given
z-scores.
For a given z-score, the probability can be
found in a table in the back of the text
(Table IV), also see inside front cover.
Note: The table only gives the areas under
the curve to the right between 0 and z. To
find other intervals requires some tricks.
Table 5.1
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Find probability z is between 1.33 and +1.33.
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Want probability z is between -1.33 and +1.33.
Solution: Locate 1.33 in the row labeled 1.3 and the column
labeled .03. By symmetry, ans = 2(0.4082) = .8164
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Find probability z exceeds 1.96 in absolute
value.
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Areas under the standard normal curve for
z exceeding 1.96 in absolute value
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Areas under the standard normal curve for
z exceeding 1.96 in absolute value
Revelation!
It follows that the area of the unshaded region is 0.95. Agrees with
Empirical Rule which states that,
for data sets having a mound
shaped distribution, 95% of the
values lie within approximately 2
standard deviations of the mean
Keys to success
Learn the standard normal table and how to
use it.
We will be using these tables through out
the course.