Stat 155, Section 2, Last Time Pepsi Challenge: When are results “significant” vs. “random”?...

Stat 155, Section 2, Last Time• Pepsi Challenge:

When are results “significant” vs. “random”?

• Independence– Conditional Prob’s = Unconditional Prob’s– Special case of and rule (of probability)

• Random Variables– Discrete vs. Continuous– Discrete:

• Summarize probability with table• Sum entries to calculate prob’s

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 277-286, 291-305

Approximate Reading for Next Class:

Pages 291-305, 334-351

Midterm I

Coming up: Tuesday, Feb. 27

Material: HW Assignments 1 – 6

Extra Office Hours:

Mon. Feb. 26, 8:30 – 12:00, 2:00 – 3:30

(Instead of Review Session)

Bring Along:

1 8.5” x 11” sheet of paper with formulas

Midterm I

Suggestions for studying:

• Exam based on HW, not text or class

• Constructed by modifying HW problems

• So rework HW problems

Note: different from “looking over HW”

Midterm I

Use Posted Old Exams:

On Class Web Page

• Least effective: read over solutions

• Moderate: read test, think, then look

• Best: Rework, then check

Midterm I

Warning about Old Exams:

• Have slightly less material

• In particular probability not covered then

• Because of different calendar

(drop date used to be earlier)

• Maybe something we haven’t covered

• Clarify by email…

Midterm I

Warning about Old Exams:

• Famous last words….

“But I knew everything on the practice exam”

• Practice exam only about “method” of questions

• Not representative of material

• Only a sample

• As present exam will be

Random Variables

Now consider continuous random variables

Recall: for measurements (not counting)

Model for continuous random variables:

Calculate probabilities as areas,

under “probability density curve”, f(x)

Continuous Random Variables

Model probabilities for continuous random

variables, as areas under “probability

density curve”, f(x):

= Area( )

a b

(calculus notation)

bXaP

b

a

dxxf )(

Continuous Random Variables

Note:

Same idea as “idealized distributions” above

Recall discussion from:

Page 8, of Class Notes, Jan. 23

Continuous Random Variables

e.g. Uniform Distribution

Idea: choose random number from [0,1]

Use constant density: f(x) = C

Models “equally likely”

To choose C, want: Area

1 = P{X in [0,1]} = C

So want C = 1. 0 1

Uniform Random Variable

HW:

4.54 (0.73, 0, 0.73, 0.2, 0.5)

4.56 (1, ½, 1/8)

Continuous Random Variables

e.g. Normal Distribution

Idea: Draw at random from a normal

population

f(x) is the normal curve (studied above)

Review some earlier concepts:

Normal Curve Mathematics

The “normal density curve” is:

usual “function” of

circle constant = 3.14…

natural number =

2.7…

,2

21

21

)(

x

exf

x

Normal Curve Mathematics

Main Ideas:

• Basic shape is:

• “Shifted to mu”:

• “Scaled by sigma”:

• Make Total Area = 1: divide by

• as , but never

2

21x

e

2

0

221 x

e2

21

x

e

0)( xf x

Computation of Normal Areas

EXCEL

Computation:

works in terms of

“lower areas”

E.g. for

Area < 1.3

)5.0,1(N

Computation of Normal Probs

EXCEL Computation:

probs given by “lower

areas”

E.g. for X ~ N(1,0.5)

P{X < 1.3} = 0.73

Normal Random Variables

As above, compute probabilities as areas,

In EXCEL, use NORMDIST & NORMINV

E.g. above: X ~ N(1,0.5)

P{X < 1.3} =NORMDIST(1.3,1,0.5,TRUE)

= 0.73 (as in pic above)

Normal Random Variables

HW:

4.57, 4.58 (0.965, ~0)

And now for something completely different

Recall

Distribution

of majors of

students in

this course:

Stat 155, Section 2, Majors

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Busine

ss /

Man

.

Biolog

y

Public

Poli

cy /

Health

Pharm

/ Nur

sing

Jour

nalis

m /

Comm

.

Env. S

ci.

Other

Undec

ided

Fre

qu

ency

And now for something completely different

A photographer for a national magazine was assigned to get photos of a great forest fire.

Smoke at the scene was too thick to get any good shots, so he frantically called his home office to hire a plane.

"It will be waiting for you at the airport!" he was assured by his editor.

And now for something completely different

As soon as he got to the small, rural airport, sure enough, a plane was warming up near the runway.

He jumped in with his equipment and yelled, "Let's go! Let's go!"

The pilot swung the plane into the wind and soon they were in the air.

And now for something completely different

"Fly over the north side of the fire," said the photographer, "and make three or four low level passes.“

"Why?" asked the pilot.

"Because I'm going to take pictures! I'm a photographer, and photographers take pictures!" said the photographer with great exasperation.

And now for something completely different

After a long pause the pilot said, "You mean you're not the instructor?“ …

Means and Variances

(of random variables) Text, Sec. 4.4

Idea: Above population summaries, extended

from populations to probability distributions

Connection: frequentist view

Make repeated draws,

from the distribution

nXXX ,...,, 21

Discrete Prob. Distributions

Recall table summary of distribution:

Taken on by random variable X,

Probabilities: P{X = xi} = pi

(note: big difference between X and

x!)

Values x1 x2 … xk

Prob. p1 p2 … pk

Discrete Prob. Distributions

Table summary of distribution:

Recall power of this:

Can compute any prob., by summing pi

Values x1 x2 … xk

Prob. p1 p2 … pk

Mean of Discrete Distributions

Frequentist approach to mean:

kkii x

nxX

xnxX ##

11

n

XXX n1

i

k

iikk xpxpxp

111

n

xxXxxX kkii ## 11

Mean of Discrete Distributions

Frequentist approach to mean:

a weighted average of values

where weights are probabilities

i

k

iixpX

1

Mean of Discrete Distributions

E.g. Above Die Rolling Game:

Mean of distribution =

= (1/3)(9) + (1/6)(0) +(1/2)(-4) = 3 - 2 = 1

Interpretation: on average (over large number

of plays) winnings per play = $1

Conclusion: should be very happy to play

Winning 9 -4 0

Prob. 1/3 1/2 1/6

Mean of Discrete Distributions

Terminology: mean is also called:

“Expected Value”

E.g. in above game “expect” $1 (per play)

(caution: on average over many plays)

Expected Value

HW:

4.59, 4.61

Expected Value

An application of Expected Value:

Assess “fairness” of games (e.g. gambling)

Major Caution: Expected Value is not what is

expected on one play, but instead is

average over many plays.

Cannot say what happens in one or a few

plays, only in long run average

Expected Value

E.g. Suppose have $5000, and need $10,000

(e.g. you owe mafia $5000, clean out safe at work. If you give to mafia, you go to jail, so decide to try to raise additional $5000 by gambling)

And can make even bets, where P{win} = 0.48

(can really do this, e.g. bets on Red in roulette at a casino)

Expected Value

E.g. Suppose have $5000, and need $10,000 and can make even bets, w/ P{win} = 0.48

Pressing Practical Problem:

• Should you make one large bet?

• Or many small bets?

• Or something in between?

Expected Value

E.g. Suppose have $5000, and need $10,000 and can make even bets, w/ P{win} = 0.48

Expected Value analysis:

E(Winnings) = P{lose} x $0 + P{win} x $2

= 0.52 x $0 + 0.48 x $2 =

= $0.96

Thus expect to lose $0.04 for every dollar bet

Expected ValueE.g. Suppose have $5000, and need $10,000

and can make even bets, w/ P{win} = 0.48

Expect to lose $0.04 for every dollar bet

• This is why gambling is very profitable

(for the casinos, been to Las Vegas?)

• They play many times

• So expected value works for them

• And after many bets, you will surely lose

• So should make fewer, not more bets?

Expected ValueE.g. Suppose have $5000, and need $10,000

and can make even bets, w/ P{win} = 0.48

Another view:

Strategy P{get $10,000}

one $5000 bet 0.48 ~ 1/2

two $2500 bets ~ (0.48)2 ~ 1/4

four $1250 bets ~ (0.48)2 ~ 1/16

“many” “no chance”

Expected ValueE.g. Suppose have $5000, and need $10,000

and can make even bets, w/ P{win} = 0.48

Surprising (?) answer:

• Best to make one big bet

• Not much fun…

• But best chance at winning

Casino Folklore:

• This really happens

• Folks walk in, place one huge bet….

Expected Value

Warning about Expected Value:

Excellent for some things, but not all decisions

e.g. if will play many times

e.g. if only play once

(so don’t have long run)

Expected ValueReal life decisions against Expected Value:

1. State Lotteries– State sells tickets– Keeps about half of $$$– Gives rest to ~ one (randomly chosen) player– So Expected Value is clearly negative– Why do people play? Totally irrational?– Players buy faint hope of humongous gain– Could be worth joy of thinking about it

Expected ValueReal life decisions against Expected Value:

1. State Lotteries– Support ours in North Carolina?

Interesting (and deep) philosophical balances:– Only totally voluntary tax– Yet tax burden borne mostly by poor– Is that fair?– Otherwise lose revenue to other states…

Expected ValueReal life decisions against Expected Value:

2. Casino Gambling– Always lose in long run (expected value…)– Yet people do it. Are they nuts?– Depends on how many times they play– If really enjoy being ahead sometimes– Then could be worth price paid for the thrill– Serious societal challenge:

(some are totally consumed by thrill)

Expected ValueReal life decisions against Expected Value:3. Insurance

– Everyone pays about 2 x Expected Loss– Insurance Company keeps the rest!– So very profitable.– But e.g. car insurance is required by law!– Sensible, since if lose, can lose very big– Yet purchase is totally against Expected Value– OK, since you only play once (not many times)– Insurance Co’s play many times (Expected

Value works for them)– So they are an evening out mechanism

And now for something completely different

A Fun Movie Clip:

405 Landing – Video File

Functions of Expected ValueImportant Properties of the Mean:i. Linearity:

Why?

i. e. mean “preserves linear transformations”

i i i

iiiiibaX bpxapbaxp

ba XbaX

bapbxpa Xi

ii

ii

Functions of Expected Value

Important Properties of the Mean:

ii. summability:

Why is harder, so won’t do here

i. e. can add means to get mean of sums

i. e. mean “preserves sums”

YXYX

Functions of Expected Value

E. g. above game:

If we “double the stakes”, then want:

“mean of 2X”

Recall $1 before

i.e. have twice the expected value

Winning 9 -4 0

Prob. 1/3 1/2 1/6

2$22 XX

Functions of Expected ValueE. g. above game:

If we “play twice”, then have

Same as above?

But isn’t playing twice different from doubling

stake?

Yes, but not in means

Winning 9 -4 0

Prob. 1/3 1/2 1/6

2$1$1$2121

XXXX

Functions of Expected ValueHW:

4.73

Indep. Of Random Variables

Independence: Random Variables X & Y

are independent when knowledge of

value of X does not change chances of

values of Y

Indep. Of Random Variables

HW:

4.71, 4.72 (Indep., Dep., Dep.)

IndependenceApplication: Law of Large Numbers

IF are independent draws from the

same distribution, with mean ,

THEN:

(needs more mathematics to make precise,

but this is the main idea)

nXX ,...,1

X

n"lim"

IndependenceApplication: Law of Large Numbers

Note: this is the foundation of the

“frequentist view of probability”

Underlying thought experiment is based on

many replications, so limit works….

Variance of Random Variables

Again consider discrete random variables:

Where distribution is summarized by a table,

Values x1 x2 … xk

Prob. p1 p2 … pk

Variance of Random Variables

Again connect via frequentist approach:

n

iin XX

nXX

1

21 1

1,...,var

1

222

21

nXXXXXX n

1## 2

111

nXxxXXxxX kii

Variance of Random Variables

Again connect via frequentist approach:

2211 XxpXxp kk

n

iin XX

nXX

1

21 1

1,...,var

22

11

1#

1#

Xxn

xXXx

nxX

kkii

k

iii Xxp

1

2

Variance of Random VariablesSo define:

Variance of a distribution

As:

random variable

k

jXjjX xp

1

22

Variance of Random Variables

E. g. above game:

=(1/2)*5^2+(1/6)*1^2+(1/3)*8^2

Note: one acceptable Excel form,

e.g. for exam (but there are many)

Winning 9 -4 0

Prob. 1/3 1/2 1/6

2222 1931

1061

1421 X

X

Standard Deviation

Recall standard deviation is square root of

variance (same units as data)

E. g. above game:

Standard Deviation

=sqrt((1/2)*5^2+(1/6)*1^2+(1/3)*8^2)

Winning 9 -4 0

Prob. 1/3 1/2 1/6

Variance of Random VariablesHW:

C16: Find the variance and standard

deviation of the distribution in 4.59.

(1.21, 1.10)

Properties of Variancei. Linear transformation

I.e. “ignore shifts” var( ) = var

( )

(makes sense)

And scales come through squared

(recall s.d. on scale of data, var is square)

222XbaX a

Properties of Variance

ii. For X and Y independent (important!)

I. e. Variance of sum is sum of variances

Here is where variance is “more natural”

than standard deviation:

222YXYX

22YXX

Properties of Variance

E. g. above game:

Recall “double the stakes”, gave same mean, as “play twice”, but seems different

Doubling:

Play twice, independently:

Note: playing more reduces uncertainty

(var quantifies this idea, will do more later)

Winning 9 -4 0

Prob. 1/3 1/2 1/6

222 4 XX

2222 22121 XXXXX

Variance of Random Variables

HW:

C17: Suppose that the random variable X

models winter daily maximum

temperatures, and that X has mean 5o C

and standard deviation 10o C.

(a) Let Y be the temp. in degrees Fahrenheit

What is the mean of X? (41o)

Hint: Recall the conversion: C=(5/9)(F-32)