the binomial, poisson, and normal distributions modified after powerpoint by carlos j....
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The Binomial, The Binomial, Poisson, and Poisson, and
Normal Normal DistributionsDistributions
Modified after PowerPoint by Carlos J. Rosas-Modified after PowerPoint by Carlos J. Rosas-AndersonAnderson
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Probability distributionsProbability distributions
We use probability We use probability distributions distributions because they work because they work –they fit lots of –they fit lots of data in real worlddata in real world
Ht (cm) 1996
100
80
60
40
20
0
Std. Dev = 14.76
Mean = 35.3
N = 713.00
Height (cm) of Hypericum cumulicola at Archbold Biological Station
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Random variableRandom variable
The mathematical rule (or function) The mathematical rule (or function) that assigns a given numerical value that assigns a given numerical value to each possible outcome of an to each possible outcome of an experiment in the sample space of experiment in the sample space of interest.interest.
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Random variables Random variables
Discrete random variablesDiscrete random variables
Continuous random variablesContinuous random variables
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The Binomial DistributionThe Binomial DistributionBernoulli Random VariablesBernoulli Random Variables
Imagine a simple trial with only two possible Imagine a simple trial with only two possible outcomesoutcomes Success (Success (SS)) Failure (Failure (FF))
ExamplesExamples Toss of a coin (heads or tails)Toss of a coin (heads or tails) Sex of a newborn (male or female)Sex of a newborn (male or female) Survival of an organism in a region (live or Survival of an organism in a region (live or
die)die)
Jacob Bernoulli (1654-1705)
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The Binomial DistributionThe Binomial DistributionOverviewOverview
Suppose that the probability of success is Suppose that the probability of success is pp
What is the probability of failure?What is the probability of failure? qq = 1 – = 1 – pp
ExamplesExamples Toss of a coin (Toss of a coin (SS = head): = head): pp = 0.5 = 0.5 qq = 0.5 = 0.5 Roll of a die (Roll of a die (SS = 1): = 1): pp = 0.1667 = 0.1667 qq = 0.8333 = 0.8333 Fertility of a chicken egg (Fertility of a chicken egg (SS = fertile): = fertile): pp = 0.8 = 0.8
qq = 0.2 = 0.2
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The Binomial DistributionThe Binomial DistributionOverviewOverview
Imagine that a trial is repeated Imagine that a trial is repeated nn times times
ExamplesExamples A coin is tossed 5 timesA coin is tossed 5 times A die is rolled 25 timesA die is rolled 25 times 50 chicken eggs are examined50 chicken eggs are examined
Assume Assume pp remains constant from trial to trial and remains constant from trial to trial and that the trials are statistically independent of that the trials are statistically independent of each othereach other
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The Binomial DistributionThe Binomial DistributionOverviewOverview
What is the probability of obtaining What is the probability of obtaining xx successes successes in in nn trials? trials?
ExampleExample What is the probability of obtaining 2 heads What is the probability of obtaining 2 heads
from a coin that was tossed 5 times?from a coin that was tossed 5 times?
PP((HHTTTHHTTT) = (1/2)) = (1/2)55 = 1/32 = 1/32
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The Binomial DistributionThe Binomial DistributionOverviewOverview
But there are more possibilities:But there are more possibilities:
HHTTTHHTTT HTHTTHTHTT HTTHTHTTHT HTTTHHTTTH
THHTTTHHTT THTHTTHTHT THTTHTHTTH
TTHHTTTHHT TTHTHTTHTH
TTTHHTTTHH
PP(2 heads) = 10 × 1/32 = 10/32(2 heads) = 10 × 1/32 = 10/32
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The Binomial DistributionThe Binomial DistributionOverviewOverview
In general, if trials result in a series of success In general, if trials result in a series of success and failures,and failures,
FFSFFFFSFSFSSFFFFFSF…FFSFFFFSFSFSSFFFFFSF…
Then the probability of Then the probability of xx successes successes in that orderin that order isis
PP((xx)) = = qq qq pp qq = = ppxx q qnn – – xx
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The Binomial DistributionThe Binomial DistributionOverviewOverview
However, if order is not important, thenHowever, if order is not important, then
where is the number of ways to where is the number of ways to obtain obtain xx successes successes
in in nn trials, and trials, and ii! = ! = ii ( (ii – 1) – 1) ( (ii – 2) – 2) … … 2 2 1 1
n!
x!(n – x)!ppxx q qn – xn – xPP((xx) =) =
n!
x!(n – x)!
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The Binomial DistributionThe Binomial DistributionOverviewOverview
Bin(0.1, 5)
0
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0.8
0 1 2 3 4 5
Bin(0.3, 5)
0
0.1
0.2
0.3
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0 1 2 3 4 5
Bin(0.5, 5)
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5
Bin(0.7, 5)
0
0.1
0.2
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0 1 2 3 4 5
Bin(0.9, 5)
0
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0 1 2 3 4 5
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The Poisson DistributionThe Poisson DistributionOverviewOverview
When there is a large number When there is a large number of trials, but a small probability of trials, but a small probability of success, binomial calculation of success, binomial calculation becomes impracticalbecomes impractical Example: Number of deaths Example: Number of deaths
from horse kicks in the Army from horse kicks in the Army in different yearsin different years
The mean number of successes The mean number of successes from from nn trials is trials is µµ = = npnp Example: 64 deaths in 20 Example: 64 deaths in 20
years from thousands of years from thousands of soldierssoldiers
Simeon D. Poisson (1781-1840)
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The Poisson DistributionThe Poisson DistributionOverviewOverview
If we substitute If we substitute µµ//nn for for pp, and let , and let nn tend to tend to infinity, the binomial distribution becomes the infinity, the binomial distribution becomes the Poisson distribution:Poisson distribution:
P(x) =e -µµx
x!
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The Poisson DistributionThe Poisson DistributionOverviewOverview
Poisson distribution is applied where random Poisson distribution is applied where random events in space or time are expected to occurevents in space or time are expected to occur
Deviation from Poisson distribution may indicate Deviation from Poisson distribution may indicate some degree of non-randomness in the events some degree of non-randomness in the events under studyunder study
Investigation of cause may be of interestInvestigation of cause may be of interest
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The Poisson DistributionThe Poisson DistributionEmission of Emission of -particles-particles
Rutherford, Geiger, and Bateman (1910) counted Rutherford, Geiger, and Bateman (1910) counted the number of the number of -particles emitted by a film of -particles emitted by a film of polonium in 2608 successive intervals of one-polonium in 2608 successive intervals of one-eighth of a minuteeighth of a minute What is What is nn?? What is What is pp??
Do their data follow a Poisson distribution?Do their data follow a Poisson distribution?
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The Poisson DistributionThe Poisson DistributionEmission of Emission of -particles-particles
No. No. --particlesparticles
ObserveObservedd
00 575711 20320322 38338333 52552544 53253255 40840866 27327377 13913988 454599 2727
1010 10101111 441212 001313 111414 11
Over 14Over 14 00TotalTotal 26082608
Calculation of Calculation of µµ::
µµ= No. of particles per = No. of particles per intervalinterval
= 10097/2608= 10097/2608
= 3.87= 3.87
Expected values:Expected values:2680 P(x) = e -3.87(3.87)x
x!2608
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The Poisson DistributionThe Poisson DistributionEmission of Emission of -particles-particles
No. No. --particlesparticles
ObserveObservedd
ExpecteExpectedd
00 5757 545411 203203 21021022 383383 40740733 525525 52552544 532532 50850855 408408 39439466 273273 25425477 139139 14014088 4545 686899 2727 2929
1010 1010 11111111 44 441212 00 111313 11 111414 11 11
Over 14Over 14 00 00TotalTotal 26082608 26802680
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The Poisson DistributionThe Poisson DistributionEmission of Emission of -particles-particles
Random events
Regular events
Clumped events
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The Poisson DistributionThe Poisson Distribution0.1
0
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The Expected Value of a The Expected Value of a Discrete Random VariableDiscrete Random Variable
nn
n
iii papapapaXE
...)( 22111
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The Variance of a Discrete The Variance of a Discrete Random VariableRandom Variable
22 )()( XEXEX
n
i
n
iiiii paap
1
2
1
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Uniform random Uniform random variables variables
0
0.1
0.2
0 1 2 3 4 5 6 7 8 9 10
X
P(X)
The closed unit interval, which The closed unit interval, which contains all numbers between 0 and 1, contains all numbers between 0 and 1, including the two end points 0 and 1including the two end points 0 and 1
Subinterval [5,6]Subinterval [3,4]
otherwise
xxf
,0
100,10/1)(
The probability density function
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The Expected Value of a The Expected Value of a continuous Random continuous Random
VariableVariable
dxxxfXE )()(
2/)()( abXE
For an uniform random variable x, where f(x) is defined on the interval [a,b], and where a<b,
and 12
)()(
22 abX
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The Normal DistributionThe Normal DistributionOverviewOverview
Discovered in 1733 by de Moivre as an Discovered in 1733 by de Moivre as an approximation to the binomial distribution when approximation to the binomial distribution when the number of trails is largethe number of trails is large
Derived in 1809 by GaussDerived in 1809 by Gauss
Importance lies in the Central Limit Theorem, Importance lies in the Central Limit Theorem, which states that the sum of a large number of which states that the sum of a large number of independent random variables (binomial, Poisson, independent random variables (binomial, Poisson, etc.) will approximate a normal distributionetc.) will approximate a normal distribution
Example: Human height is determined by a Example: Human height is determined by a large number of factors, both genetic and large number of factors, both genetic and environmental, which are additive in their environmental, which are additive in their effects. Thus, it follows a normal distribution.effects. Thus, it follows a normal distribution. Karl F.
Gauss (1777-1855)
Abraham de Moivre (1667-1754)
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The Normal DistributionThe Normal DistributionOverviewOverview
A A continuouscontinuous random variable is said to be random variable is said to be normally distributed with mean normally distributed with mean and variance and variance 22 if if its probability its probability densitydensity function is function is
ff((xx) is not the same as ) is not the same as PP((xx)) PP((xx) would be 0 for every ) would be 0 for every xx because the normal because the normal
distribution is continuousdistribution is continuous However, However, PP((xx11 < < XX ≤ ≤ xx22) = ) = ff((xx))dxdx
f (x) =
1
2(x )2/22
e
xx11
xx22
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The Normal DistributionThe Normal DistributionOverviewOverview
0.00
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0.45
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
x
f(x)
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The Normal DistributionThe Normal DistributionOverviewOverview
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
x
f(x)
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The Normal DistributionThe Normal DistributionOverviewOverview
Mean changes Variance changes
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The Normal DistributionThe Normal DistributionLength of FishLength of Fish
A sample of rock cod in Monterey Bay suggests that A sample of rock cod in Monterey Bay suggests that the mean length of these fish is the mean length of these fish is = 30 in. and = 30 in. and 22 = 4 = 4 in.in.
Assume that the length of rock cod is a normal random Assume that the length of rock cod is a normal random variablevariable
If we catch one of these fish in Monterey Bay,If we catch one of these fish in Monterey Bay, What is the probability that it will be at least 31 in. What is the probability that it will be at least 31 in.
long?long? That it will be no more than 32 in. long?That it will be no more than 32 in. long? That its length will be between 26 and 29 inches?That its length will be between 26 and 29 inches?
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The Normal DistributionThe Normal DistributionLength of FishLength of Fish
What is the probability that it will be at least 31 What is the probability that it will be at least 31 in. long?in. long?
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25 26 27 28 29 30 31 32 33 34 35
Fish length (in.)
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The Normal DistributionThe Normal DistributionLength of FishLength of Fish
That it will be no more than 32 in. long?That it will be no more than 32 in. long?
0.00
0.05
0.10
0.15
0.20
0.25
25 26 27 28 29 30 31 32 33 34 35
Fish length (in.)
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The Normal DistributionThe Normal DistributionLength of FishLength of Fish
That its length will be between 26 and 29 inches?That its length will be between 26 and 29 inches?
0.00
0.05
0.10
0.15
0.20
0.25
25 26 27 28 29 30 31 32 33 34 35
Fish length (in.)
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-6 -4 -2 0 2 40
1000
2000
3000
4000
5000
Standard Normal Standard Normal DistributionDistribution
μμ=0 and =0 and σσ22=1=1
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Useful properties of the Useful properties of the normal distributionnormal distribution
1.1. The normal distribution has The normal distribution has useful properties:useful properties:
Can be added E(X+Y)= E(X)Can be added E(X+Y)= E(X)+E(Y) and +E(Y) and σσ2(X+Y)= 2(X+Y)= σσ2(X)+ 2(X)+ σσ2(Y)2(Y)
Can be transformed with Can be transformed with shiftshift and and change of scalechange of scale operationsoperations
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Consider two random variables Consider two random variables X and YX and Y
Let X~N(Let X~N(μμ,,σσ) ) and letand let Y=aX+b Y=aX+b where a and b where a and b area constantsarea constants
Change of scaleChange of scale is the operation of is the operation of multiplying multiplying XX by a constant “ by a constant “aa” because ” because one unit of X becomes “a” units of Y.one unit of X becomes “a” units of Y.
ShiftShift is the operation of adding a constant is the operation of adding a constant ““bb” to X because we simply move our ” to X because we simply move our random variable X “b” units along the x-random variable X “b” units along the x-axis.axis.
If X is a normal random variable, then the If X is a normal random variable, then the new random variable Y created by this new random variable Y created by this operations on X is also a random normal operations on X is also a random normal variable variable
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For X~N(For X~N(μμ,,σσ) ) and and Y=aX+bY=aX+b
E(Y) =aE(Y) =aμμ+b+b σσ22(Y)=a(Y)=a22 σσ22
A special case of a change of scale A special case of a change of scale and shift operation in which a = 1/and shift operation in which a = 1/σσ and b =-1(and b =-1(μμ//σσ))
Y=(1/Y=(1/σσ)X-)X-μμ//σσ Y=(X-Y=(X-μμ)/)/σσ gives gives E(Y)=0 and E(Y)=0 and σσ22(Y) =1(Y) =1
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The Central Limit The Central Limit TheoremTheorem
That Standardizing any random variable That Standardizing any random variable that itself is a sum or average of a set of that itself is a sum or average of a set of independent random variables results independent random variables results in a new random variable that is nearly in a new random variable that is nearly the same as a standard normal one.the same as a standard normal one.
The only caveats are that the sample The only caveats are that the sample size must be large enough and that the size must be large enough and that the observations themselves must be observations themselves must be independent and all drawn from a independent and all drawn from a distribution with common expectation distribution with common expectation and variance.and variance.
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Exercise Exercise
Location of the measurement