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Unformatted text preview: 1 Chapter 8 Continuous Probability Distributions 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 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 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 ) ( 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 ) ( ) ( 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 ) ( 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 ) ( 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. 9 1. pr(X = k) = 0 for all numbers k. • Hence: f" of Height Average " × ε = + ≤ < = = → }) ({ lim ) ( ε ε k X k pr k X pr f(k) = × = + ≤ < }) ({ ε k X k pr 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 = + ≤ < → ε ε ε k X k pr ) ( ) ( }) ({ k F k F k X k pr + = + ≤ < ε ε f(k) ) ( ) ( lim ) ( ' = + = → ε ε ε k F k F k F 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 ) ( ) ( 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 ) ( + ≤ < = → x X x pr x f 13 • A concrete (counter)example: – Sometimes f takes on values greater than 1....
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This note was uploaded on 01/20/2011 for the course ECON 15A taught by Professor Shirey during the Winter '08 term at UC Irvine.
 Winter '08
 Shirey

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