Exercise 2 Theorem 2. If X has the Beta distribution as given by Equation (12), thenE[X] =a+B(13)ProofWriteE[X] = af(z) dz =r(a+ B) ,I(@)r(3)la+1)-1(1 – 2)8-1 dx =r(a+1) I(@+B)r(a) r(a+B+1) J.T(a + 3 +1),r(a+1)r(8)2la+1)-1(1 – x)8-! dr =T(a +1) T(@+ B)(1)T(a) T(a+B+1)Using the recursion relation for the gamma function:a + BSimilarly,Theorem 3. If X has theeta distribution as given by Equation (12), thea(a+1)(a+B+1)(a+ B)E[X² =(14)%3DExercise 2. Prove Theorem 3.

Answered: Image /qna-images/answer/178ce8e8-1650-46ee-849f-7cb183d4677d.jpg

Theorem 2. If X has the Beta distribution as given by Equation (12), then E[X] = a +B (13) %3D Proof Write E[X] = | xf(x) d = r(a+6) „(a+1)-1(1 – 2)8-1 dx = r(a)r(3) T(a+B+1) „(a+1)–1(1 – x)8–1 dx = Г(о+ 1) Г() r(a+1) T(a+ B) T(a) I(a+B+1). T(a +1) T(@+ B) T(@) T(a+B+ 1) * Using the recursion relation for the gamma function: a + B Similarly, Theorem 3. If X has the Beta distribution as given by Equation (12), then a(a +1) E[X²] = (a+B+1)(a+ B) (14) Exercise 2. Prove Theorem 3.

Theorem 2. If X has the Beta distribution as given by Equation (12), then E[X] = a +B (13) %3D Proof Write E[X] = | xf(x) d = r(a+6) „(a+1)-1(1 – 2)8-1 dx = r(a)r(3) T(a+B+1) „(a+1)–1(1 – x)8–1 dx = Г(о+ 1) Г() r(a+1) T(a+ B) T(a) I(a+B+1). T(a +1) T(@+ B) T(@) T(a+B+ 1) * Using the recursion relation for the gamma function: a + B Similarly, Theorem 3. If X has the Beta distribution as given by Equation (12), then a(a +1) E[X²] = (a+B+1)(a+ B) (14) Exercise 2. Prove Theorem 3.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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