Transcript Lecture11

Problems
Problems 4.62, 4.64, 4.66, 4.68
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.
5. Continuous Random Variables
There are two types of continuous distributions
we discuss now: uniform and normal
distributions.
5. Continuous Random Variables
There are two types of continuous distributions
we discuss now: uniform and normal
distributions.
A density curve is the graph of a continuous
probability distribution.
5. Continuous Random Variables
There are two types of continuous distributions
we discuss now: uniform and normal
distributions.
A density curve is the graph of a continuous
probability distribution.
1) This curve always positive (or 0)
2) The area under the curve is 1.
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
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
Example
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
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.
What we can do is find probabilities over
intervals?
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.
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. In general
this can be found by looking at the area
under this curve between x=1 and x=4.
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.
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

2
d c

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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
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
We represent a standard normal variable
with a z instead of an x.
Convert any normal distribution to a
standard normal distribution by using the
z-score.
z 
x

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
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.
Examples
Use the tables in the back of the book to find
the following.
a) P(0  z  2.43)
b) P(-2.43  z  0)
c) P(1.20  z  2.30)
d) P(-1.50  z  2.4)
e) P( z  1.8)
Problems
Problems 5.3, 5.4, 5.12
Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40,
5.48
Keys to success
Learn the standard normal table and how to
use it.
We will be using these tables through out
the course.
Homework
• Review Chapter 5.1-5.2
• Read Chapters 5.3, 5.4, 6.1, 6.2
• Assignment due Friday at noon
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