7Theoretical Probability Distributions

Random variables (RV)

Represented by X,Y, or ZDiscrete or continuous RVDiscrete RV martial status: single, married, divorcedContinuous RV weight, height

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7.1 Probability distributions

Every RV has a corresponding probability distribution.

X = the birth order of each child born to a woman residing in USX = 1, 2 first-born, second-born child

Let X = the RV, x = the outcome of a particular child

P(X=4) = 0.058P(X=1 or X=2) = 0.416 + 0.330 = 0.746

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7.1 Probability distributions

Probability distribution of Table 7.1 data.

Probabilities that are calculated (from a finite amount of data, based on theoretical consideration) are called (empirical, theoretical) probabilities.

7.2 The binomial distribution

Dichotomous RV, Y = life and death, male and female, sickness and healthAlso known as Bernoulli RVExample Y denotes smoking status, Y=0,1 non-smoking, smoking In 1987, 29% of the adults in the US smoked cigar, cigarettes or pipesP(Y=1) = p = 0.29 P(Y=0) = 0.71 X denotes the number of persons selected from the population of adults in the US X can take on three possible values: 0, 1, 2

P(X=0) = (1-p)(1-p) = 0.504P(X=1) = p(1-p) + (1-p)p = 0.412P(X=2) = p*p = 0.084

P(X=0) + P(X=1) + P(X=2) = 1

7.2 The binomial distribution

X would be a binomial RV with parameters n=3 and p=0.29P(X=0) = (1-p) (1-p) (1-p) = 0.358P(X=1) = p(1-p) (1-p) + (1-p)p (1-p) + (1-p) (1-p)p = 0.439P(X=2) = p*p (1-p) + p (1-p)p + (1-p)p*p = 0.179P(X=3) = p*p*p = 0.024

In case X=n

(mean,variance) of X = (np, np(1-p))For n=10, (np, np(1-p)) = (10*0.29, 10*0.29*(0.71) = (2.9, 2.059)

xnxnx ppCxXP )1()(

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=

=

Skew to right

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==

symmetric

7.3 The Poisson distribution

When n>>1, and p is very small, such as p = the probability of a person involved in a motor vehicle accident each year in the US = 0.00024

The Poisson distribution is used to model disctete events that occur infrequently in time or space.

X is said to have a Poisson distribution with parameter !

)(x

exXP

x

7.3 The Poisson distribution

Binomial distribution, np, np(1-p), if p <<1 np, np mean = varianceExample Determine the number of people in a population of 10000 who will be involved in a motor vehicle accident per year = 10000*0.00024 = 2.4

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218.0!1

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24.2

14.2

04.2

eXP

eXP

eXP

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The Poisson distribution is highly skewed for small , as increases, the distribution becomes more symmetric.

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The Poisson distribution is highly skewed for small , as increases, the distribution becomes more symmetric.

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7.4 The Normal distribution

Discrete binomial or Poisson distribution as n increases Normal distribution

where -∞ ＜ x ＜ ∞

2

2

1

2

1)(

x

exf

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7.4 The Normal distribution

Change of variable standard normal distribution

With mean =0, variance 2= 1

2

2

1

2

1)(

zezf

xZ

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=- =

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Figure 7.10 The standard normal curve, area between z = -2.00 and z = 2.00

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=

Normal distribution table

NORMDIST - Area under the curve start from left hand side

Z=0

Z=2

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5.0

2

XZ

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Let X = systolic blood pressure. For the population of 18- to 74-year-old males in the U

S, systolic 收縮的 blood pressure is distributed with a mean 129 mm Hg and standard deviation 19.8 mm Hg.

Find the value of x that cuts off the upper 2.5% of the curve of systolic blood pressure,Find P(X>x) = 0.025

for the upper 2.5% z = 1.96 = (x – 129)/ 19.8 x = 167.8 mm Hg

Symmetric (the lower 2.5%)z = -1.96 x = 90.2 mm Hg

Comparison of two normal distributions (ND)

Not taking corrective medication, diastolic 舒張 blood pressure is approximately ND with mean = 80.7 mm Hg, s.d = 9.2 mm HgFor the men using antihypertensive drugs, with mean = 94.9 mm Hg, s.d = 11.5 mm Hg

ExampleIdentify 90% of the persons who are currently taking medication, what value of diastolic blood pressure should be designated as the lower cutoff point ?From Table, lower 10% z = -1.28 x = 80.2 mm HgBelow 80.2 mm Hg represent FNPerson who is taking medication are not identified as such

Other probability distributionsNegative binomial distribution, multi-nomial distribution, hypergeometric distribution

Negative binomial distributionWhen X=x, among the previous x-1 test, r-1 times are success, x-r times are failure

ExampleA telegraph system has a probability of 0.1sending wrong message. What is the probability that the 10th message is the third error ?

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Multi-nomial distribution

n independent tests, each test has r types of outcome, where each type has a probability of occurrence p1, ….., pr. Let the RV be X=(X1, ….Xr).

Example A dice is thrown 10 times, what is the probabilities that number 1,3 and 5 occur 2,3,and 5 times respectively ?

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Hypergeometric distribution

N balls, R red color balls, N-R white color balls, RV, X = n balls are drawn without replacement X is said to have hypergeometric distribution - the probability of having x red ball from R red balls, and n-x white ball for N-R white balls.

ExampleA cargo of 50 goods, 5 are defected and 45 are good. Five pieces are drawn, what is the probability of identify defected goods ?

P(X 1) = 1 – P(X 0) = 1-f(0)≧ ≦

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7.5 Further applications

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