Transcript Linear Transformation and Statistical Estimation and the Law of

Linear Transformation and Statistical
Estimation and the Law of Large
Numbers
Target Goal:
I can describe the effects of transforming a
random variable.
I can calculate expected winnings.
6.2a
h.w: pg 356: 27 – 30, 37, 39 -41, 43, 45
Mean and Variance for
Continuous Random Variables
For continuous probability distributions, mx and sx can
be defined and computed using methods from calculus.
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The mean value mx locates the center of the
continuous distribution.
The standard deviation, sx, measures the
extent to which the continuous distribution
spreads out around mx.
A company receives concrete of a certain type
from two different suppliers.
Let
xThe
= compression
strength of a randomly selected
first supplier is preferred to the second both in terms of
batchmean
fromvalue
Supplier
1
and variability.
y = compression strength of a randomly selected
batch from Supplier 2
Suppose that
mx = 4650 pounds/inch2 sx = 200 pounds/inch2
my = 4500 pounds/inch2 sy = 275 pounds/inch2
4300
4500
my
4700
mx
4900
What would happen to the mean and standard deviation if we had
Suppose
Wolf
Grocery
had
totalof business
to deductCity
$100 from
everyone’s
salaryabecause
being bad? are the
of 14 employees. The following
monthly salaries of all the employees.
3500
1300
1200
1500
1900
1700
1400
2300
2100
1200
1800
1400
1200
1300
The
mean
andofstandard
of the
Let’s graph
boxplots
these monthly deviation
salaries to see what
happens to the
distributions . . .
monthly
salaries are
mx = $1700 and sx = $603.56 What happened to
We
see that the distribution just shifts to the right
What
100 unitsto
but the spread is the same.
happened
the standard
deviations?
Suppose
business is really good, so the manager
the means?
gives everyone a $100 raise per month. The new
mean and standard deviation would be?
mx = $1800 and sx = $603.56
Wolf City Grocery Continued . . .
mx = $1700 and sx = $603.56
Suppose the manager gives everyone a 20% raise
graph
boxplots
of these
salaries todeviation
see what happens
to thebe
-Let’s
the
new
mean
andmonthly
standard
would
distributions . . .
mx = $2040 and sx = $724.27
Notice that multiplying by a
constant stretches the
distribution, thus, changing
Notice that both the mean and standard deviation increased by 1.2.
the standard deviation.
Mean and Standard Deviation of
Linear functions
If x is a random variable with mean, mx,
and standard deviation, sx, and a and b
are numerical constants, and the random
variable y is defined by y  a  bx
and
m y  ma bx  a  bm x
s s
2
y
2
a  bx
b s
2
2
x
or s y  b s x
Consider the chance experiment in which a customer of
a propane gas company is randomly selected. Let x be
the number of gallons required to fill a propane tank.
Suppose that the mean and standard deviation is 318
gallons and 42 gallons, respectively. The company is
considering the pricing model of a service charge of
$50 plus $1.80 per gallon.Let y be the random variable
of the amount billed.
What is the equation for y? y = 50 + 1.8x
What are the mean and standard deviation for the amount
billed?
my = 50 + 1.8(318) = $622.40
sy = 1.8(42) = $75.60
Linear Transformations
How does multiplying or dividing by a
constant affect a random variable?
Effect on a Random Variable of Multiplying (Dividing) by a Constant
Multiplying (or dividing) each value of a random variable by a number b:
•
Multiplies (divides) measures of center and location (mean, median,
quartiles, percentiles) by b.
•
Multiplies (divides) measures of spread (range, IQR, standard deviation)
by |b|.
•
Multiplying does not change the shape of the distribution.
Note: Multiplying a random variable by a
constant b multiplies the variance by b2.
Transforming and Combining Random Variables
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Linear Transformations
Effect on a Random Variable of Adding (or Subtracting) a Constant
Adding the same number a (which could be negative) to each value of a
random variable:
•
Adds a to measures of center and location (mean, median, quartiles,
percentiles).
•
Adding does not change measures of spread (range, IQR, standard
deviation).
•
Does not change the shape of the distribution.
Transforming and Combining Random Variables
How does adding or subtracting a constant
affect a random variable?
Summary: Linear transformations
 Shape:
same as the probability
distribution of X.
 Center: m y  a  bm x
 Spread: s Y  b s X
Recall:
Statistics obtained from probability
samples are random variables
because their values would vary in
repeated sampling.
x is an estimate for μ.
 If we choose a different random sample,
the luck of the draw will probably
produce a different x
Law of Large Numbers
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Draw independent observations at random
from any population with finite mean μ.
As the number of observations increases,
the sample mean approaches mean μ of
the population.
The more variation in the outcomes, the
more trials are needed to ensure that is
close to μ.
The law of large numbers in action
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Suppose the mean is 64.5.
What do you notice about the distribution?
As we increase the size of our sample, the
`always approaches the mean μ of the
population!
What type of businesses use law of
large numbers for pricing ect?
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Casinos
Insurance
Fast food restaurants
Averaging over many individuals
produces stable results.
Law of Small Numbers
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Write down a sequence of heads and tails
for 10 tosses.
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What was the longest run of heads or tails?
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What do you think the probability of a run of
3 or more heads or tails in 10 is?
The probability is > 0.80!
Most people incorrectly think that short
sequences of random events show us the
kind of behavior that appears only in the
long run.
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The probability of both a run of 3 heads and
3 tails is almost 0.2.
Most sequences in the short run don’t
seem random to us.
In the long run, they “even” out to the
expected mean.
Example: The “Hot Hand in
Basketball”
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If a basketball player makes several
consecutive shots, what do you believe?
She has the hot hand and should make the
next shot? or,
She is “due” to miss?
Neither:
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Random behavior says each shot is
independent of the previous shot.
Over the long run the regular behavior
described by probability and law of large
numbers takes over.
Example: casinos – the mean guarantees
the house a profit.
Exercise: A Game of Chance
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The law of large numbers says that if we
take enough outcomes, their average value
is sure to approach the mean of the
distribution.
Would you take this wager … ?
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Toss a coin 10 times
If there is no run of three or more heads or
tails in the 10 outcomes, I’ll pay you $2.
If there is a run of 3 or more, you pay me
$1.
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Simulate enough plays of the game to
estimate the mean outcome.
The outcomes are +$2 if you win and -$1 if
you lose.
Is it to your advantage to play ?
To simulate: open list & spreadsheets page
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In “=“ cell enter
Menu:Data:Random:randint(1,2,10)
Tally and record L, the length of the longest
run of heads or tails per trail.
Do 20 trials (place cursor in the formula cell
and press enter twice to generate new
trial).
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Calculate your mean winnings.
P(W) = P(1 or 2 in a row, not 3 in a row)
P(L) = 1 – P(W)
Expected outcome: $2P(W) + -$1P(L)
Fill in table and report back.
What is your expected outcome?
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The actual expected outcome is:
μ = ($2)((0.1738) + (-$1)(0.8262)
= -$0.4785
On the average you would lose about 48
cents each time you play.
How close we come to this depends on
how many trails.