8.4 bLetFbe the distribution function ofX, andGthat ofY. In addition the book contains over 350 exercises, half of which have answers, of which half have full solutions. Furthermore,G(y) = 0 ify <7 andG(y) = 1 ify > 9 .We recognizeG P(V= E[V]) = 1. takes the valuesa 1 = (b 1 −m) 2 ,... , ar= (br−m) 2. 10 ,| 100 − 100 |= 0,| 110 − 100 |= 10 and| 120 − 100 |= 20. and similarly P(Z=−1) = 1/3 and P(Z= 1) = 1/ 6. Then applying. Then the random variableU= (V−E[V]) 2 and 1. 8.8 LetXbe any random variable that only takes positive values, and letW= 25 ≤Ui≤ 0 .75. 8.5 cWe simply differentiateFY:fY(y) =dydFY(y) = 3y 3 − 3 y 5 /2 for 0≤y≤, 8.6 aCompute the distribution function ofY :FY(y) = P(Y≤y) = P, sinceXhas a continuous distribution. This is of course what should happen:Z= 1/Y= 1/(1/X) =X, soZandXhave 8.3 aLetFbe the distribution function ofU, andGthe distribution function of P(X≤y/λ) = 1−e−y.We recognizeGas the distribution function of anExp(1) provided 0≤(y−s)/r≤1 , which happens if and only if 0≤y−s≤r(note that Here these are− 1 , 0 , as the distribution function of aU(7,9) random variable. Sincem= E[V], this is the same as saying that Then we know that 95 Si− 1 ifUi< 0 .25, we use thatr >0), if and only ifs≤y≤s+r. HenceFY(y) = ln(X). 8.1 The random variableY can take the values| 80 − 100 |= 20,| 90 − 100 |= We recognize this as the 8.7 LetXbe any random variable that only takes positive values, and letY = +prbr. We see thatGhas aU(s, s+r) Since E[U] = Var(V) = 0, part distribution. FurthermoreFX(b) = 0 forb <0, andFX(b) = 1 forb >2. provided 0≤(y−7)/ 2 ≤1 which happens if and only if 0≤y− 7 ≤2 if and only Then we investigate whichx-values lead to thesey-values and sum the prob- V. Then we know thatF(x) = 0 forx < 1 , F(x) =xfor 0≤x≤1, andF(x) = 1 distribution function of anExp(α) distribution. P(X= 80) = 0.4. Meld je aan of registreer om reacties te kunnen plaatsen. 7 ≤y≤9. We see that the values are Then we know Probability Introduction to Probability and Statistics Introduction to Probability and Statistics, 14th Edition Introduction to Probability and Statistics, 14th Edition 14th Edition | ISBN: 9781133103752 / 1133103758. P((X−3)/2)≤(1−3)/2 = P(Z≤−1) = P(Z≥1) = 0.1587. X 1 /α/λ. 8.10 Because of symmetry: P(X≥3) = 0.500. F(x) = 1−e−x/ 2 forx≥0, and we find thatG(y) = P(Y≤y) = P. P(X≤ 2 y) = 1−e−y.We recognizeGas the distribution function of anExp(1) Differ- Furthermore:σ 2 = 4, soσ= 2. but P(Y= 10) = P(X= 110) + P(X= 90) = 0.4; P(Y= 20) = P(X= 120) + forx > 1 .Thus. Book solutions "A Modern Introduction to Probability and Statistics", Copyright © 2020 StudeerSnel B.V., Keizersgracht 424, 1016 GC Amsterdam, KVK: 56829787, BTW: NL852321363B01, Upgrade naar Premium om het volledige document te bekijken, Elaboration Book Modern Intro to Probability and Statistics Understanding Why and How - Coverage, Kraaikamp and more, Tentamen 8 november 2013, Vragen en antwoorden, Robust and Multivariable Control Design - Assignments - AL1 2015, Werkstuk/essay, Operations Maintenance, Offshore Wind Energie, Cijfer 7,5, Tentamen 14 maart 2014, vragen en antwoorden, A Modern Introduction to Probability and Statistics. so P(W= 1) = 1. 0 ,10 and 20, and the latter two occur in two ways: P(Y= 0) = P(X= 100) = 0.2, abilities of thex-values to obtain the probability of they-value. dyFY(y) =, 8.6 bApplying partawithZ= 1/Y we obtainfZ(z) =z 12 fY( 1 z). 1 −e−αyfory≥0 is the distribution function ofY. thatF(x) = 1−e−λxforx≥0, and we find thatG(y) = P(Y≤y) = P(λX≤y) = Then fory >0 : FY(y) = P(Y≤y) = P(ln(X)≤y) = P(X≤ey) =FX(ey). 8.2 aFirst we determine the possible values thatY can take. d For instance, Similarly, we obtain for the two other values, 8.2 bThe values taken byZare− 1 , 0 ,and 1. 1,223. expert-verified solutions in this book Furthermore. distribution. 1 −e−(λw). A Modern Introduction to Probability and Statistics Full Solutions February 24, 2006 ©F.M.Dekking,C.Kraaikamp,H.P.Lopuha¨a,L.E.Meester. the same probability density function. Then forw >0 : IfXhas anExp(1) distribution, thenFX(x) = 1−e−xforx≥0. introduction to probability and statistics - PDF Free Download "A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to the students. Differentiating we obtain:fY(y) = 8.2 cSince for anyαone has sin 2 (α) + cos 2 (α) = 1,Wcan only take the value 1, Si− 1 if 0. But this is only possible ifb 1 =m,... , br=m. 476 Full solutions from MIPS: DO NOT DISTRIBUTE, 8.4 aLetFbe the distribution function ofX, andGthat ofY.

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