Normal Binomial Distribution
-
Upload
yasheshgaglani -
Category
Documents
-
view
237 -
download
0
Transcript of Normal Binomial Distribution
-
8/4/2019 Normal Binomial Distribution
1/41
Psyc 235:
Introduction toStatistics
DONT FORGET TO SIGN IN FOR CREDIT!
http://www.psych.uiuc.edu/~jrfinley/p235/
-
8/4/2019 Normal Binomial Distribution
2/41
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
-
8/4/2019 Normal Binomial Distribution
3/41
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
-
8/4/2019 Normal Binomial Distribution
4/41
Random Variables
Example:
Random Experiment:
flip a coin 3 times
Random Variable:# of heads
-
8/4/2019 Normal Binomial Distribution
5/41
Random Variables
Discrete vs Continuous finite vs infinite # possible outcomes
Scales of MeasurementCategorical/Nominal
Ordinal
IntervalRatio
-
8/4/2019 Normal Binomial Distribution
6/41
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, ...)
-
8/4/2019 Normal Binomial Distribution
7/41
So far...
Graphing & summarizing sampledistributions (DESCRIPTIVE)
Counting Rules Probability
Random Variables
one more key concept is needed to startdoing INFERENTIAL statistics:
SAMPLING DISTRIBUTION
-
8/4/2019 Normal Binomial Distribution
8/41
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
-
8/4/2019 Normal Binomial Distribution
9/41
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
-
8/4/2019 Normal Binomial Distribution
10/41
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 = ....
-
8/4/2019 Normal Binomial Distribution
11/41
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!
-
8/4/2019 Normal Binomial Distribution
12/41
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
-
8/4/2019 Normal Binomial Distribution
13/41
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 = ....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
Sampling Distribution
Bi i l Si i
-
8/4/2019 Normal Binomial Distribution
14/41
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 = 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
Sampling Distribution
-
8/4/2019 Normal Binomial Distribution
15/41
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
-
8/4/2019 Normal Binomial Distribution
16/41
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
Sampling Distribution
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
-
8/4/2019 Normal Binomial Distribution
17/41
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
-
8/4/2019 Normal Binomial Distribution
18/41
Lets see some moreBinomial Distributions
What happens if we try doing adifferent # of trials (n) ?
That is, try a different sample size...
-
8/4/2019 Normal Binomial Distribution
19/41
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
-
8/4/2019 Normal Binomial Distribution
20/41
Binomial Distribution, p=.5, n=10
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
-
8/4/2019 Normal Binomial Distribution
21/41
Binomial Distribution, p=.5, n=20
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
-
8/4/2019 Normal Binomial Distribution
22/41
Binomial Distribution, p=.5, n=50
0
0.02
0.04
0.06
0.08
0.1
0.12
# of successes
-
8/4/2019 Normal Binomial Distribution
23/41
Binomial Distribution, p=.5, n=100
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
# of successes
-
8/4/2019 Normal Binomial Distribution
24/41
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...
-
8/4/2019 Normal Binomial Distribution
25/41
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
-
8/4/2019 Normal Binomial Distribution
26/41
Binomial Distribution, p=.1, n=5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5
# of successes
-
8/4/2019 Normal Binomial Distribution
27/41
Binomial Distribution, p=.1, n=10
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
-
8/4/2019 Normal Binomial Distribution
28/41
Binomial Distribution, p=.1, n=20
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
-
8/4/2019 Normal Binomial Distribution
29/41
Binomial Distribution, p=.1, n=50
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
# of successes
-
8/4/2019 Normal Binomial Distribution
30/41
Binomial Distribution, p=.1, n=100
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
# of successes
N l A i ti
-
8/4/2019 Normal Binomial Distribution
31/41
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)
-
8/4/2019 Normal Binomial Distribution
32/41
0
0.05
0.1
0.15
0.2
0.25
0.3
-
8/4/2019 Normal Binomial Distribution
33/41
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:
-
8/4/2019 Normal Binomial Distribution
34/41
different
means
different
standard
deviations
-
8/4/2019 Normal Binomial Distribution
35/41
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
-
8/4/2019 Normal Binomial Distribution
36/41
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
-
8/4/2019 Normal Binomial Distribution
37/41
Standard NormalDistribution
has mean: =0
has standard deviation: =1 ANY Normal Distribution can be
converted to the Standard Normal
Distribution...
-
8/4/2019 Normal Binomial Distribution
38/41
StandardNormal
Distribution
Normal Distributions &
-
8/4/2019 Normal Binomial Distribution
39/41
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
-
8/4/2019 Normal Binomial Distribution
40/41
Standard NormalDistribution
So, if this were a Sampling Distribution, ...
-
8/4/2019 Normal Binomial Distribution
41/41
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