By John D. Enderle, David C. Farden, Daniel J. Krause

This can be the second one in a sequence of 3 brief books on likelihood conception and random strategies for biomedical engineers. This quantity makes a speciality of expectation, ordinary deviation, moments, and the attribute functionality. moreover, conditional expectation, conditional moments and the conditional attribute functionality also are mentioned. together disbursed random variables are defined, besides joint expectation, joint moments, and the joint attribute functionality. Convolution can be constructed. a substantial attempt has been made to improve the idea in a logical manner—developing distinctive mathematical abilities as wanted. The mathematical history required of the reader is simple wisdom of differential calculus. each attempt has been made to be in step with ordinary notation and terminology—both in the engineering neighborhood in addition to the likelihood and facts literature. the purpose is to arrange scholars for the appliance of this concept to a large choice of difficulties, besides supply practising engineers and researchers a device to pursue those issues at a extra complex point. Pertinent biomedical engineering examples are used in the course of the textual content.

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**Intermediate Probability Theory for Biomedical Engineers**

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4. The joint PDF for the RVs x and y is ⎧ ⎨ β 2 , f x,y (α, β) = α ⎩ 0, 0<β≤ √ α<1 elsewhere. 25), (c) whether or not x and y are independent random variables. Answers: 1, ln (4), no. 5. With the joint PDF of random variables x and y given by f x,y (α, β) = aα 2 β, 0, 0 ≤ α ≤ 3, 0 ≤ β ≤ 1 otherwise, where a is a constant, determine: (a) a, (b)P (0 ≤ x ≤ 1, 0 ≤ y ≤ 1/2), (c )P (xy ≤ 1), (d )P (x + y ≤ 1). Answers: 1/108, 7/27, 2/9, 1/270. 6. 25). Answers: 13/16, 49/256, 5/4, 40. 2 BIVARIATE RIEMANN-STIELTJES INTEGRAL The Riemann-Stieltjes integral provides a unified framework for treating continuous, discrete, and mixed RVs—all with one kind of integration.

Then 1 f x (α) = lim T→∞ 2π T φx (t)e − j αt d t. 46) −T Proof. The desired result follows from the above theorem by letting b = α, a = α − h, and h > 0. Then f x (α) = lim h→0 Fx (α) − Fx (α − h) h 1 = lim T→∞ 2π T e j ht − 1 − j αt e φx (t) d t. h→0 j th lim −T In some applications, a closed form for the characteristic function is available but the inversion integrals for obtaining either the CDF or the PDF cannot be obtained analytically. In these cases, a numerical integration may be performed efficiently by making use of the FFT (fast Fourier transform) algorithm.

Let RV x have mean ηx and variance σx2 . (a) Show that E(|x − a|2 ) = σx2 + (ηx − a)2 for any real constant a. (b) Find a so that E(|x − a|2 ) is minimized. 13. The random variable y has η y = 10 and σ y2 = 2. Find (a) E(y 2 ) and (b) E((y − 3)2 ). 14. 5. 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.