Transcript Notes on Linear Transformations

Warm Up
• How do I know this is a probability
distribution?
• What is the probability that Mary hits
x=# red
lights
p(x)
0
0.05
1
0.25
2
0.35
3
0.15
4
0.15
5
0.05
exactly 3 red lights?
• What is the probability that she gets
at least 4 red lights?
• What is the probability that she gets
less than two?
• Find the mean & standard deviation.
Find Mean & Standard Deviation:
x=#
books
read
P(x)
0
0.13
1
0.21
2
0.28
3
0.31
4
0.07
Ex.
1.
2.
3.
x = possible
winnings
P(x)
5
0.1
7
0.31
8
0.24
10
0.16
14
0.19
Find the mean
Find the Standard Deviation
Find the probability that x is within one
deviation from the mean.
LINEAR
TRANSFORMATIONS
Section 6.2A
Remember – effects of Linear
Transformations
• Adding or Subtracting a Constant
• Adds “a” to measures of center and location
• Does not change shape or measures of spread
• Multiplying or Dividing by a Constant
• Multiplies or divides measures of center and location by “b”
• Multiplies or divides measures of spread by |b|
• Does not change shape of distribution
Adding/Subtracting a constant from data shifts the mean but
doesn’t change the variance or standard deviation.
•
•
E  X  c   E( X )  c
Var  X  c   Var ( X )
Multiplying/Dividing by a constant multiplies the mean and
the standard deviation.
E (aX )  aE ( X )
𝜎𝑥 aX = a ∙ 𝜎𝑥
Var (aX )  a Var ( X )
2
Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The
vehicle will hold up to 6 passengers. The number of passengers X
on a randomly selected day has the following probability
distribution. He charges $150 per passenger. How much on average
does Pete earn from the half-day trip?
# Passengers
Prob
2
0.15
3
0.25
4
0.35
5
0.2
6
0.05
Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The
vehicle will hold up to 6 passengers. The number of passengers X
on a randomly selected day has the following probability
distribution. He charges $150 per passenger. What is the typical
deviation in the amount that Pete makes?
# Passengers
Prob
2
0.15
3
0.25
4
0.35
5
0.2
6
0.05
What if it costs Pete $100 to buy permits, gas, and a ferry pass for
each half-day trip. The amount of profit V that Pete makes from the
trip is the total amount of money C that he collects from the
passengers minus $100. That is V = C – 100. So, what is the average
profit that Pete makes? What is the standard deviation in profits?
A large auto dealership keeps track of sales made during each hour
of the day. Let X = the number of cars sold during the first hour of
business on a randomly selected Friday. Based on previous records,
the probability distribution of X is shown below. Suppose the
dealership’s manager receives a $500 bonus from the company for
each car sold. What is the mean and standard deviation of the
amount that the manager earns on average?
# cars sold
Prob
0
0.3
1
0.4
2
0.2
3
0.1
Suppose the dealership’s manager receives a $500 bonus
from the company for each car sold. To encourage customers
to buy cars on Friday mornings, the manager spends $75 to
provide coffee and doughnuts. Find the mean and standard
deviation of the profit the manager makes.
# cars sold
Prob
0
0.3
1
0.4
2
0.2
3
0.1
Variance of y = a + bx
• Relates to slope.
y
b
x
2

y
b2  2
x
2
2
2
y  b  x
var( y )  b  var( x ) 
2
 y2  b 2 x2
Effects of Linear Transformation on the
Mean and Standard Deviation if 𝒀 =
𝒂 + 𝒃𝑿.
𝜇𝑦 = 𝑎 + 𝑏𝜇𝑥
𝜎𝑦 = 𝑏 𝜎𝑥
*Shape remains the same.
Example: Three different roads feed into a freeway entrance. The number of
cars coming from each road onto the freeway is a random variable with mean
values as follows. What’s the mean number of cars entering the freeway.
Mean #
Road
Cars
1
800
2
1000
3
600
Mean of the Sum of Random Variables
For any two random variables, X and Y, if 𝑇 = 𝑥 + 𝑦
then the expected value of T is
𝐸 𝑇 = 𝜇 𝑇 = 𝜇𝑥 + 𝜇𝑦
Ex: What is the standard deviation of the # of cars coming
from each road onto the freeway.
Road
Mean #
Cars
St.
Dev.
1
800
34.5
2
1000
42.8
3
600
19.3
Variance of the Sum of Random
Variables
For any two random variables, X and Y, if 𝑇 = 𝑥 + 𝑦
then the variance of T is
x
y
 x y 
3 x  2 y 
Mean
20
24
st dev
5
3
x
y
 x y 
3x  y 
Mean
20
24
st dev
5
3
x
y
 x y 
 3x2 y 
Mean
20
24
st dev
5
3
x
y
 x y 
 3x  y 
Mean
st dev
20
24
5
3
Find: 
and 
3 x2 y
3 x2 y
x
P(x)
y
P(y)
3
0.32
10
0.22
4
0.14
20
0.34
5
0.12
30
0.18
6
0.42
40
0.26
Homework
Worksheet