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Chap08 - Chapter 8 Continuous Probability Distributions 1 1...

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1 Chapter 8 Continuous Probability Distributions
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2 1. Continuous distributions 1. pdf: Probability Density Function 2. cdf: Cumulative Distribution Function 3. Probability 4. More on pdfs 2. Uniform Distribution 3. Expectations 4. Normal Distribution 1. Central Limit Theorem
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3 Continuous Probability Distributions We often want to explore the distribution of quantities that vary continuously Height, weight, IQ Quarterly profits, income/expenditure ratios Probability of a sudden acceleration/brake failure
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4 A function f is a probability density function (pdf) if: 1. f(x) ≥ 0, for every number x; 2. where: 1 ) ( = +∞ - dx x f + - +∞ - = a a a dx x f dx x f ) ( lim ) (
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5 If f is the pdf of a continuous distribution, then F is the cumulative distribution function (cdf) of that distribution: - = x dx x f x F ) ( ) (
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6 We define the probability of the event of the random variable yielding a value less than (or equal to) some number a as: Pr(X ≤ a) = F(a) Similarly, the probability of X being greater than a is: Pr(X > a) = 1 – F(a) - = a dx x f ) ( - - = a dx x f ) ( 1 +∞ = a dx x f ) (
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7 We define the probability of the event of the random variable yielding a value in the interval (a, b]: Pr(a < X ≤ b) = F(b) – F(a) - - - = a b dx x f dx x f ) ( ) ( = b a dx x f ) (
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8 3 important features of the pdf f*: 1. Our construction of f ensures that pr(X = c) = 0, for any number c. 2. f is a derivative (of the cdf F) . 3. f is not a probability function. *These features hold for the distributions used in this course; in general matters are more complex.
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9 1. pr(X = k) = 0 for all numbers k. Hence: f" of Height Average " × ε = + < = = }) ({ lim ) ( 0 ε ε k X k pr k X pr 0 f(k) 0 = × = + < }) ({ ε k X k pr
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10 2. f is a derivative (of the cdf F) So: So: But: So: f" of Height Average " }) ({ × = + < ε ε k X k pr f" of Height Average " }) ({ = + < ε ε k X k pr f(k) }) ({ lim 0 = + < ε ε ε k X k pr ) ( ) ( }) ({ k F k F k X k pr - + = + < ε ε f(k) ) ( ) ( lim ) ( ' 0 = - + = ε ε ε k F k F k F
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11 There is another way that we can tell that f is a derivative From the Fundamental Theorem of Calculus, we have the relationship: ) ( ) ( x f x F dx d = - = x dx x f x F ) ( ) (
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12 3. f is not a probability function. Notice that pdfs take single numbers as their arguments, probability functions take sets of numbers as their arguments. f gives the curve that distributes the probability along the real line ε ε ε }) ({ lim ) ( 0 + < = x X x pr x f
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13 A concrete (counter-)example: Sometimes f takes on values greater than 1. Probability functions, by definition, cannot do this! = otherwise 0, 1 0 if , 2 ) ( x x x f 5 . 1 ) 75 (. = f 1 ) ( 1 0 = dx x f
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14 The uniform distribution is useful when a quantity is known/assumed to fall within a definite finite interval, and there is no further information. E.g., without checking the schedule, you arrive at a subway stop, knowing only that the subway arrives every 20 minutes.
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