Epidemiology 9509 probability dist’ns (continued)

Epidemiology 9509Principle of Biostatistics

Chapter 5Probability Distributions (continued)

John Koval

Department of Epidemiology and BiostatisticsUniversity of Western Ontario

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Epidemiology 9509 probability dist’ns (continued)

What was covered previously

1. probability P(A)setsP(A and B); P(A or B)

2. probability distributions2.1 discrete

2.1.1 equiprobable

2.1.2 bernoulli

2.1.3 binomial

2.1.4 poisson

2.2 continuous

2.2.1 uniform

2.2.2 normal

3. calculating probabilities

3.1 discretePr(X = x)

3.2 continuousintervals: Pr(X < a), Pr(a < X < b)

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Epidemiology 9509 probability dist’ns (continued)

What is being covered now

Using SAS to

1. calculate probabilities

2. calculate and plot probability distributions

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Epidemiology 9509 probability dist’ns (continued)

Calculating probabilities

SAS function PDF

title ’calculate binomial probability’;

data binom1;

prob = pdf(’binomial’, 4, 0.4, 10);

output ;

proc print data=binom1;

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Epidemiology 9509 probability dist’ns (continued)

binomial probability

calculate binomial probability

Obs prob

1 0.25082

Does this agree with previous calculations?

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Epidemiology 9509 probability dist’ns (continued)

binomial probability

calculate binomial probability

Obs prob

1 0.25082

Does this agree with previous calculations?0.251, Lecture Chapter 5, page 8

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Epidemiology 9509 probability dist’ns (continued)

Calculating probability distribution

title "calculate binomial probability distribution’;

data binom2;

do x = 0 to 10 by 1;

prob = pdf(’binomial’, x, 0.4, 10);

output;

end;

proc print data=binom2;

proc gplot;

plot prob*x;

run;

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Epidemiology 9509 probability dist’ns (continued)

binomial probability distribution

calculate binomial probability distribution

Obs x prob

1 0 0.00605

2 1 0.04031

3 2 0.12093

4 3 0.21499

5 4 0.25082

6 5 0.20066

7 6 0.11148

8 7 0.04247

9 8 0.01062

10 9 0.00157

11 10 0.00010

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Epidemiology 9509 probability dist’ns (continued)

GPLOT of pdf

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Epidemiology 9509 probability dist’ns (continued)

Calculating cumulative probabilities

values up to and includingSAS function CDF

title ’calculate cumulative binomial probability’;

data binom3;

prob = cdf(’binomial’, 7, 0.4, 20);

output ;

proc print data=binom3;

run;

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate cumulative binomial probability

Obs prob

1 0.41589

Does this agree with previous calculations ?

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate cumulative binomial probability

Obs prob

1 0.41589

Does this agree with previous calculations ?0.4159, using R, Lecture Chapter 5, page 30

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Epidemiology 9509 probability dist’ns (continued)

cumulative continuous probabilities

Pr(X < b)

= Pr

(

ZN <b−µ

σ

)

= Φ(

b−µ

σ

)

Φ() given by SAS function PROBNORM

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Epidemiology 9509 probability dist’ns (continued)

example

Recall normal approximation to binomial

wantPr(Xnorm < 7.5)= Pr(ZN <

(

7.5−82.19

)

= Φ(−.228)

title ’calculate Normal probability’;

data norm1;

prob =probnorm(-0.228);

output;

proc print data=norm1;

run;

;

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate Normal probability

Obs prob

1 0.40982

Does this agree with previous calculations ?

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate Normal probability

Obs prob

1 0.40982

Does this agree with previous calculations ?0.4098by linear interpolation,see lecture Chapter 5, page 30

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Epidemiology 9509 probability dist’ns (continued)

Probability of interval

Pr(17 < X < 22)= Pr

(

17−205 < ZN <

22−205

)

= Pr(−0.6 < ZN < 0.4) = Φ(0.4) − Φ(−0.6)

title ’calculate Normal probability for interval’;

data norm2;

a=-0.6;

b=0.4;

proba =probnorm(a);

probb = probnorm(b);

probint = probb - proba;

output;

proc print data=norm2;

run;

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate Normal probability for interval

Obs a b proba probb probint

1 -0.6 0.4 0.27425 0.65542 0.38117

Does this agree with previous calculations ?

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Epidemiology 9509 probability dist’ns (continued)

binomial cumulative distribution calculation

calculate Normal probability for interval

Obs a b proba probb probint

1 -0.6 0.4 0.27425 0.65542 0.38117

Does this agree with previous calculations ?0.3809,see lecture Chapter 5, page 26

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Epidemiology 9509 probability dist’ns (continued)

Plotting normal density function

not usually done in practice

data norm3;

do x = 0 to 10 by 0.05;

density = pdf(’normal’, x, 4, 1.55);

output ;

end;

proc gplot data = norm3;

plot density*x;

symbol interpol=join;

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Epidemiology 9509 probability dist’ns (continued)

GPLOT of pdf of Normal N(4,2.4)

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Epidemiology 9509 probability dist’ns (continued)

normal approximation to binomial

title ’Normal approximation to binomial’;

data normbinom;

n=20;

pi=0.4;

mu = n*pi;

var = n*pi*(1-pi);

sd = sqrt(var);

do i = 0 to 20.975 by 0.025;

binompdf = pdf(’binomial’, floor(i), pi, n);

x = i-0.5;

normpdf = pdf(’normal’, x, mu, sd);

output normbinom;

end;

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Epidemiology 9509 probability dist’ns (continued)

normal approximation to binomial(continued)

proc gplot data=normbinom;

plot binompdf * x normpdf * x/

haxis=-1 to 21 by 1 vaxis=0 to 0.2 by 0.05 overlay;

symbol interpol=join;

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Epidemiology 9509 probability dist’ns (continued)

GPLOT of normal approximation to Bin(20,0.4)

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Epidemiology 9509 probability dist’ns (continued)

another normal approximation to binomial

non-symmetric distribution Bin(10,.2)

data normbinom2;

n=10;

pi=0.2;

mu = n*pi;

var = n*pi*(1-pi);

sd = sqrt(var);

do i = 0 to 10.9075 by 0.025;

binompdf = pdf(’binomial’, floor(i), pi, n);

x = i-0.5;

normpdf = pdf(’normal’, x, mu, sd);

output normbinom2;

end;

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Epidemiology 9509 probability dist’ns (continued)

non-symmetric distribution (continued)

proc gplot data=normbinom2;

plot binompdf * x normpdf * x /

haxis=-1 to 11 by 1 vaxis=0 to 0.5 by 0.05 overlay;

symbol interpol=join;

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Epidemiology 9509 probability dist’ns (continued)

Normal approximation to Bin(10,0.2)

original distribution is asymmetricnot a good fit to the normal

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