7 Theoretical Probability Distributions Random variables RV Represented
7 Theoretical Probability Distributions
Random variables (RV) Represented by X, Y, or Z Discrete or continuous RV Discrete RV martial status: single, married, divorced Continuous RV weight, height
7. 1 Probability distributions Every RV has a corresponding probability distribution. X = the birth order of each child born to a woman residing in US X = 1, 2 first-born, second-born child Let X = the RV, x = the outcome of a particular child P(X=4) = 0. 058 P(X=1 or X=2) = 0. 416 + 0. 330 = 0. 746 Chapter 7 p 163
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. Chapter 7 p 164
7. 2 The binomial distribution Dichotomous RV, Y = life and death, male and female, sickness and health Also known as Bernoulli RV Example Y denotes smoking status, Y=0, 1 non-smoking, smoking In 1987, 29% of the adults in the US smoked cigar, cigarettes or pipes P(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) = 0. 504 P(X=1) = p(1 -p) + (1 -p)p = 0. 412 P(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. 29 P(X=0) = (1 -p) = 0. 358 P(X=1) = p(1 -p) + (1 -p)p = 0. 439 P(X=2) = p*p (1 -p) + p (1 -p)p + (1 -p)p*p = 0. 179 P(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)
Skew to right = = Chapter 7 p 171
symmetric = = Chapter 7 p 172
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 l
7. 3 The Poisson distribution Binomial distribution, np(1 -p), if p <<1 np, np mean = variance Example Determine the number of people in a population of 10000 who will be involved in a motor vehicle accident per year l = 10000*0. 00024 = 2. 4
=l Chapter 7 p 175
The Poisson distribution is highly skewed for small l, as l increases, the distribution becomes more symmetric. Chapter 7 p 175
The Poisson distribution is highly skewed for small l, as l increases, the distribution becomes more symmetric. Chapter 7 p 175
7. 4 The Normal distribution Discrete binomial or Poisson distribution as n increases Normal distribution where -∞<x< ∞ = = Chapter 7 p 177
7. 4 The Normal distribution Change of variable standard normal distribution With mean m=0, variance s 2= 1 Chapter 7 p 177
= - Chapter 7 p 179 =
Figure 7. 10 The standard normal curve, area between z = -2. 00 and z = 2. 00 Chapter 7 p 180
= Chapter 7 p 181
Normal distribution table
NORMDIST - Area under the curve start from left hand side Z=0 Z=2
Chapter 7 p 181
Let X = systolic blood pressure. For the population of 18 - to 74 -year-old males in the US, 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 Chapter 7 p 182
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 Hg For the men using antihypertensive drugs, with mean = 94. 9 mm Hg, s. d = 11. 5 mm Hg Example Identify 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 Hg Below 80. 2 mm Hg represent FN Person who is taking medication are not identified as such
Other probability distributions Negative binomial distribution, multi-nomial distribution, hypergeometric distribution Negative binomial distribution When X=x, among the previous x-1 test, r-1 times are success, x-r times are failure Example A telegraph system has a probability of 0. 1 sending wrong message. What is the probability that the 10 th message is the third error ?
Multi-nomial distribution n independent tests, each test has r types of outcome, where each type has a probability of occurrence p 1, …. . , pr. Let the RV be X=(X 1, …. 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 ?
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. Example A 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)
7. 5 Further applications Chapter 7 p 189
Chapter 7 p 190
Chapter 7 p 171
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