By Howard Raiffa
"In the sphere of statistical selection concept, Raiffa and Schlaifer have sought to boost new analytic strategies in which the fashionable idea of software and subjective chance can truly be utilized to the industrial research of standard sampling problems."
—From the foreword to their vintage paintings Applied Statistical determination Theory. First released within the Nineteen Sixties via Harvard college and MIT Press, the booklet is now provided in a brand new paperback version from Wiley
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Let x be a RV with median m. (a) Show that for any real constant a: m E(|x − a|) = E(|x − m|) + 2 (α − a) d Fx (α). a (b) Find the constant a for which E(|x − a|) is minimized. cls 28 QC: IML/FFX T1: IML October 27, 2006 7:20 INTERMEDIATE PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS 15. Use integration by parts to show that ∞ 0 (1 − Fx (α)) d α − E(x) = Fx (α) d α. −∞ 0 16. Show that ∞ 0 Fx (α) d α + E(|x|) = −∞ (1 − Fx (α)) d α. 0 17. Random variable x has ηx = 50, σx = 5, and an otherwise unknown CDF.
Consider a department in which all of its graduate students range in age from 22 to 28. Additionally, it is three times as likely a student’s age is from 22 to 24 as from 25 to 28. Assume equal probabilities within each age group. Let random variable x equal the age of a graduate student in this department. Determine: (a) E(x), (b) E(x 2 ), (c) σx . 3. A class contains five students of about equal ability. The probability a student obtains an A is 1/5, a B is 2/5, and a C is 2/5. Let random variable x equal the number of students who earn an A in the class.
6. 5− ). 5. 5. For 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1 we have β α Fx,y (α, β) = 4α β d α dβ = α 2 β 2 . 25) 3 = . 5 P (A) = 4αβ d αdβ = 3 . 25 0 Two-dimensional representations for the PDF and CDF are shown in Fig. 7. (b) We have ⎧ 1 ⎨ f x,y (α, β)dβ = 2α, 0 ≤ α ≤ 1 f x (α) = 0 ⎩ 0, otherwise. (c) We have f y (β) = ⎧ ⎨ ⎩ 1 f x,y (α, β)dβ = 2α, 0≤α≤1 0 0, otherwise. cls T1: IML October 27, 2006 7:23 BIVARIATE RANDOM VARIABLES 47 (d) Since f x,y (α, β) = f x (α) f y (β) for all real α and β we find that the RVs x and y are independent.