How do I know which distribution to use?. Three discrete distributions Proportional Binomial...
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Transcript of How do I know which distribution to use?. Three discrete distributions Proportional Binomial...
How do I know which distribution to use?
Three discrete distributions
Proportional
Binomial
Poisson
Proportional
Binomial
Poisson
Given a number of categories Probability proportional to number of opportunitiesDays of the week, months of the year
Number of successes in n trialsHave to know n, p under the null hypothesisPunnett square, many p=0.5 examples
Number of events in interval of space or timen not fixed, not given pCar wrecks, flowers in a field
Proportional
Binomial
Poisson
Binomial with large n, small p convergesto the Poisson distribution
Examples: name that distribution
• Asteroids hitting the moon per year• Babies born at night vs. during the day
• Number of males in classes with 25 students
• Number of snails in 1x1 m quadrats• Number of wins out of 50 in rock-paper-scissors
Proportional
Binomial
Poisson
Generate expected values
Calculate
2 test statistic
Sample
Test statistic
Null hypothesis
Null distributioncompare
How unusual is this test statistic?
P < 0.05 P > 0.05
Reject Ho Fail to reject Ho
Sample
Chi-squaredTest statistic
Null hypothesis:Data fit a particular
Discrete distribution
Null distribution:2 With
N-1-p d.f.
compare
How unusual is this test statistic?
P < 0.05 P > 0.05
Reject Ho Fail to reject Ho
Chi-squared goodness of fit test
Calculate expected values
The Normal Distribution
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Babies are “normal”
Babies are “normal”
Normal distribution
-2 -1 0 1 2 3
0.1
0.2
0.3
0.4
Normal distribution
• A continuous probability distribution
• Describes a bell-shaped curve• Good approximation for many biological variables
Continuous Probability Distribution
-2 -1 0 1 2 3
0.1
0.2
0.3
0.4
The normal distribution is very
common in nature
Human body temperature
Human birth weight Number of bristles on a Drosophila abdomen
-2 -1 0 1 2 3
0.1
0.2
0.3
0.4
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Discrete probabilitydistribution
Continuous probabilitydistribution
Poisson Normal
Pr[X=2] = 0.22 Pr[X=2] = ?
Probability that X is EXACTLY2 is very very small
-2 -1 0 1 2 3
0.1
0.2
0.3
0.4
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Discrete probabilitydistribution
Continuous probabilitydistribution
Poisson Normal
Pr[X=2] = 0.22 Pr[1.5≤X≤2.5] =
Area under the curve
-2 -1 0 1 2 3
0.1
0.2
0.3
0.4
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Discrete probabilitydistribution
Continuous probabilitydistribution
Poisson Normal
Pr[X=2] = 0.22 Pr[1.5≤X≤2.5] =
0.06
Normal distribution
€
f x( ) =1
2πσ 2e−x−μ( )
2
2σ 2
-2 -1 0 1 2 3
0.1
0.2
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Normal distribution
x-2 -1 0 1 2 3
0.1
0.2
0.3
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Area under the curve:
€
Pr[a ≤ X ≤ b] = f [X]dXa
b
∫
€
Pr[a ≤ X ≤ b] = f [X]dXa
b
∫
=1
2πσ 2e−x−μ( )
2
2σ 2
a
b
∫ dX
But don’t worry about this for now!
A normal distribution is fully described by its mean and variance
Figure 10.3. The normal distribution with different values of themean and variance. (a) μ =5, σ =4; (b) μ= -3; σ=1/2.
A normal distribution is symmetric around
its mean
Y
ProbabilityDensity
About 2/3 of random draws from a normal distribution are
within one standard deviation of the mean
About 95% of random draws from a normal distribution are
within two standard deviations of the mean
(Really, it’s 1.96 SD.)
Properties of a Normal Distribution
• Fully described by its mean and variance
• Symmetric around its mean• Mean = median = mode• 2/3 of randomly-drawn observations fall between μ-σ and μ+σ
• 95% of randomly-drawn observations fall between μ-2σ and μ+2σ
Standard normal distribution
• A normal distribution with:
• Mean of zero. (μ = 0)• Standard deviation of one. (σ = 1)
-2 -1 0 1 2 3
0.1
0.2
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0.4
Standard normal table
• Gives the probability of getting a random draw from a standard normal distribution greater than a given value
Standard normal table: Z = 1.96
x.x0 x.x1 x.x2 .x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9
...
1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592
1.6 0.0548 0.0537 0.05262 0.05155 0.0505 0.04947 0.04846 0.04746 0.04648 0.04551
1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.0392 0.03836 0.03754 0.03673
1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938
1.9 0.02872 0.02807 0.02743 0.0268 0.02619 0.02559 0.025 0.02442 0.02385 0.0233
2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.0197 0.01923 0.01876 0.01831
2.1 0.01786 0.01743 0.017 0.01659 0.01618 0.01578 0.01539 0.015 0.01463 0.01426
2.2 0.0139 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.0116 0.0113 0.01101
Standard normal table: Z = 1.96
Pr[Z>1.96]=0.025
Normal Rules
• Pr[X < x] =1- Pr[X > x]
+ = 1 Pr[X<x] + Pr[X>x]=1
Standard normal is symmetric, so...• Pr[X > x] = Pr[X < -x]
Normal Rules
• Pr[X > x] = Pr[X < -x]
• Pr[X < x] =1- Pr[X > x]
Sample standard normal calculations
• Pr[Z > 1.09]• Pr[Z < -1.09]• Pr[Z > -1.75]• Pr[0.34 < Z < 2.52]• Pr[-1.00 < Z < 1.00]
What about other normal distributions?
• All normal distributions are shaped alike, just with different means and variances
What about other normal distributions?
• All normal distributions are shaped alike, just with different means and variances
• Any normal distribution can be converted to a standard normal distribution, by
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Z =Y − μ
σ
Z-score
€
Z =Y − μ
σ
Z tells us how many standard deviations Y is from the mean
The probability of getting a value greater than Y is the same as the probability of getting a value greater than Z from a standard normal distribution.
€
Z =Y − μ
σ
Z tells us how many standard deviations Y is from the mean
Pr[Z > z] = Pr[Y > y]
Example: British spies
MI5 says a man has to be shorter than 180.3 cm tall to be a spy.
Mean height of British men is 177.0cm, with standard deviation 7.1cm, with a normal distribution.
What proportion of British men are excluded from a career as a spy by this height criteria?
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Draw a rough sketch of the question
μ = 177.0cmσ = 7.1cmy = 180.3
Pr[Y > y]
€
Z =Y −μ
σ
Z =180.3−177.0
7.1Z = 0.46
x.x0 x.x1 x.x2 .x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9
0.0
0.5 0.49601
0.49202
0.48803
0.48405
0.48006
0.47608
0.4721 0.46812
0.46414
0.1
0.46017
0.4562 0.45224
0.44828
0.44433
0.44038
0.43644
0.43251
0.42858
0.42465
0.2
0.42074
0.41683
0.41294
0.40905
0.40517
0.40129
0.39743
0.39358
0.38974
0.38591
0.3
0.38209
0.37828
0.37448
0.3707 0.36693
0.36317
0.35942
0.35569
0.35197
0.34827
0.4
0.34458
0.3409 0.33724
0.3336 0.32997
0.32636
0.32276
0.31918
0.31561
0.31207
0.5
0.30854
0.30503
0.30153
0.29806
0.2946 0.29116
0.28774
0.28434
0.28096
0.2776
Part of the standard normal table
Pr[Z > 0.46] = 0.32276, so Pr[Y > 180.3] = 0.32276
Sample problem
MI5 says a woman has to be shorter than 172.7 cm tall to be a spy.
The mean height of women in Britain is 163.3 cm, with standard deviation 6.4 cm. What fraction of women meet the height standard for application to MI5?