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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 2
![Theorem 2. If X has the Beta distribution as given by Equation (12), then
E[X] =
a+B
(13)
Proof
Write
E[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 + B
Similarly,
Theorem 3. If X has the
eta distribution as given by Equation (12), the
a(a+1)
(a+B+1)(a+ B)
E[X² =
(14)
%3D
Exercise 2. Prove Theorem 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F178ce8e8-1650-46ee-849f-7cb183d4677d%2Fmk1zypj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 2. If X has the Beta distribution as given by Equation (12), then
E[X] =
a+B
(13)
Proof
Write
E[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 + B
Similarly,
Theorem 3. If X has the
eta distribution as given by Equation (12), the
a(a+1)
(a+B+1)(a+ B)
E[X² =
(14)
%3D
Exercise 2. Prove Theorem 3.
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