Normal Binomial Distribution

8/4/2019 Normal Binomial Distribution

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Psyc 235:

Introduction toStatistics

DONT FORGET TO SIGN IN FOR CREDIT!

http://www.psych.uiuc.edu/~jrfinley/p235/


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Independent vs.Dependent Events

Independent Events: unrelated events thatintersect at chance levels given relativeprobabilities of each event

Dependent Events: events that are relatedin some way

So... how to tell if two events areindependent or dependent? Look at the INTERSECTION: P(AB)

if P(AB) = P(A)*P(B) --> independent

if P(AB) P(A)*P(B) --> dependent


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Random Variables

Random Variable: variable that takes on a particular

numerical value based on outcome of arandom experiment

Random Experiment (aka Random Phenomenon):

trial that will result in one of severalpossible outcomes

cant predict outcome of any specific trial

can predict pattern in the LONG RUN


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Random Variables

Example:

Random Experiment:

flip a coin 3 times

Random Variable:# of heads


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Random Variables

Discrete vs Continuous finite vs infinite # possible outcomes

Scales of MeasurementCategorical/Nominal

Ordinal

IntervalRatio


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Data World vs. TheoryWorld

Theory World: Idealization of reality(idealization of what you might expectfrom a simple experiment)Theoretical probability distribution

POPULATION

parameter: a number that describes thepopulation. fixed but usually unknown

Data World: data that results from anactual simple experiment Frequency distribution

SAMPLE

statistic: a number that describes the sample(ex: mean, standard deviation, sum, ...)


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So far...

Graphing & summarizing sampledistributions (DESCRIPTIVE)

Counting Rules Probability

Random Variables

one more key concept is needed to startdoing INFERENTIAL statistics:

SAMPLING DISTRIBUTION


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Binomial Situation

Bernoulli Trial a random experiment having exactly two possible

outcomes, generically called "Success" and "Failure

probability of Success = p

probability of Failure = q = (1-p)

Heads Tails Good Robot BadRobot

Examples:

Coin toss: Success=Headsp=.5

Robot Factory:Success=Good Robot

p=.75


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Binomial Situation

Binomial Situation:n: # of Bernoulli trials

trials are independentp (probability of success) remains

constant across trials

Binomial Random Variable:X = # of the n trials that are

successes


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Binomial Situation:collect data!

Population:Outcomes ofall possible coin tosses

(for a fair coin)

Success=Heads

p=.5

Lets do 10 tosses

n=10 (sample size)

Bernoulli Trial:

one coin toss

Binomial Random

Variable:

X=# of the 10 tosses

that come up heads

(aka Sample Statistic)Sample: X = ....


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Binomial Distributionp=.5, n=10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2 3 4 5 6 7 8 9 10

# of successes

This is theSAMPLING DISTRIBUTION

of X!


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Sampling Distribution

Sampling Distribution:

Distribution of values that your sample

statistic would take on, if you kepttaking samples of the same size, fromthe same population, FOREVER

(infinitely many times).Note: this is a THEORETICAL

PROBABILITY DISTRIBUTION

i i l Si i


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(for a fair coin)

Success=Heads

p=.5

Lets do 10 tosses

n=10 (sample size)

Bernoulli Trial:

one coin toss

Binomial Random

Variable:


that come up heads

(aka Sample Statistic)Sample: X = ....3 5 6

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

# of successes


Bi i l Si i


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(for a fair coin)

Success=Heads

p=.5

Lets do 10 tosses

n=10 (sample size)

Bernoulli Trial:

one coin toss

Binomial Random

Variable:


that come up heads

(aka Sample Statistic)Sample: X = 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

# of successes



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Binomial Formula

P(X= k) = P(exactly kmany successes)

P(X= k) =n

k

p

k(1- p)n- k

Binomial

Random

Variable

specific # of

successes you

could get

n

k

=

n!

k!(n - k)!

combination

called the

Binomial Coefficient

probability

of success

probability

of failure

specific #

offailures

i in


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Binomial Formula

3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

# of successes


p(X=3) =

Remember this idea....

Hmm... what if we had gotten X=0?...

pretty unlikely outcome... fair coin?

ulatio

allpo

ssible

coin

oraf

aircoin)

p=.5

n=10

M th Bi i l


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More on the BinomialDistribution

X ~ B(n,p)

Expected Value

and Variance for X~B(n,p)mX = np

s X2 = np(1- p)

Standard Deviation : s X = np(1- p)

these are the

parameters forthe sampling

distribution of X

# heads in 5 tosses of a coin: X~B(5,1/2)

Expectation Variance Std. Dev.# heads in 5 tosses of a coin: 2.5 1.25 1.12

x:

L t


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Lets see some moreBinomial Distributions

What happens if we try doing adifferent # of trials (n) ?

That is, try a different sample size...


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Binomial Distribution, p=.5, n=5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5

# of successes


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0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

# of successes


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 1 1 1 2 1 3 1 4 15 16 1 7 1 8 1 9 2 0

# of successes


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0

0.02

0.04

0.06

0.08

0.1

0.12

# of successes


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

# of successes


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Whoah.

Anyone else notice those DISCRETEdistributions starting to look

smoother as sample size (n)increased?

Lets look at a few more binomial

distributions, this time with adifferent probability of success...


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Binomial Robot Factory

2 possible outcomes:

Good Robot

90%

Bad Robot

10%

Youd like to know about how many BAD robots youre likely to get

before placing an order... p = .10 (... success)

n = 5, 10, 20, 50, 100


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

# of successes


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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7 8 9 10

# of successes


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0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10 1 1 1 2 1 3 1 4 15 16 1 7 1 8 1 9 2 0

# of successes


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

# of successes


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

# of successes

N l A i ti


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Normal Approximationof the Binomial

If n is large, then

X ~ B(n,p) {Binomial Distribution}

can be approximated by a NORMAL DISTRIBUTION withparameters:

= np

s = np(1- p)


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0

0.05

0.1

0.15

0.2

0.25

0.3


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Normal Distributions

(aka Bell Curve)

Probability Distributions of a ContinuousRandom Variable (smooth curve!)

Class of distributions, all with the sameoverall shape

Any specific Normal Distribution ischaracterized by two parameters:

mean:

standard deviation:


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different

means

different

standard

deviations


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Standardizing

Standardizing a distribution of valuesresults in re-labeling &stretching/squishing the x-axis

useful: gets rid of units, puts alldistributions on same scale for comparison

HOWTO:

simply convert every value to a:Z SCORE:

z =x - m

s


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Standardizing

Z score:

Conceptual meaning: how many standard deviations from the mean

a given score is (in a given distribution)

Any distribution can be standardized

Especially useful for NormalDistributions...

z =x - m

s

Standard Normal


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Standard NormalDistribution

has mean: =0

has standard deviation: =1 ANY Normal Distribution can be

converted to the Standard Normal

Distribution...


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StandardNormal

Distribution

Normal Distributions &


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Normal Distributions &Probability

Probability = area under the curve intervals

cumulative probability

[draw on board]

For the Standard Normal Distribution:

These areas have already beencalculated for us (by someone else)

Standard Normal


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Standard NormalDistribution

So, if this were a Sampling Distribution, ...


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Next Time

More different types of distributionsBinomial, Normal

t, Chi-square F

And then... how will we use these todo inference?

Remember: biggest new idea todaywas:SAMPLING DISTRIBUTION

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Normal Binomial Distribution