For each of the following proof excerpts, explain as fully as possible what the mathematical statistics context is, and what steps are being taken (i.e. relate it to the content being assessed). Standard notation for functions and parameters is used throughout. NB: you do not need to complete the proof. (a) Σxen (ag(x) + bh(x)) f(x) = aΣx€N 9(x)ƒ (x) + bΣxen h(x)ƒ(x) (b) M"(t) = X(X – t)−¹ = &A(A − t)−² = 2\(\ — t)−³ and M" (0) : - = dt2 (c) P(Z² < x) = P(-√√x < Z <√√x) = Fz (√x) - Fz( - √∞) ∞ (d) Σxpq* = pqΣxq²- x=0 x=1 (e) F(2.5) = 1 - e-0.35×2.5 n (f) Σ etx x=0 pª (1 − p)n-x = pq = ∞ d dq x=0 d q = pq (1 dq 0.583 (3dp) n n-x = • Σ(") (pet)ª (1 − p)"-" = (1 − p + pe¹)n x=0 -1 (g) F(x)=√x for 0 < x < 1 and F-¹(x) = x², so X = U² (h) E(et(a+bX)) = E(eat ebtX) = eat Mx (bt) (i) [ ²* ³ x (2 − x) dx = ³ [x² - 2³²12] = ³ (4-8) = 1 (j) P(X > x) = P(0 events in [0, x]) = (Ax)⁰e-Ax 0! -λx so F(x) = 1- e-λx
For each of the following proof excerpts, explain as fully as possible what the mathematical statistics context is, and what steps are being taken (i.e. relate it to the content being assessed). Standard notation for functions and parameters is used throughout. NB: you do not need to complete the proof. (a) Σxen (ag(x) + bh(x)) f(x) = aΣx€N 9(x)ƒ (x) + bΣxen h(x)ƒ(x) (b) M"(t) = X(X – t)−¹ = &A(A − t)−² = 2\(\ — t)−³ and M" (0) : - = dt2 (c) P(Z² < x) = P(-√√x < Z <√√x) = Fz (√x) - Fz( - √∞) ∞ (d) Σxpq* = pqΣxq²- x=0 x=1 (e) F(2.5) = 1 - e-0.35×2.5 n (f) Σ etx x=0 pª (1 − p)n-x = pq = ∞ d dq x=0 d q = pq (1 dq 0.583 (3dp) n n-x = • Σ(") (pet)ª (1 − p)"-" = (1 − p + pe¹)n x=0 -1 (g) F(x)=√x for 0 < x < 1 and F-¹(x) = x², so X = U² (h) E(et(a+bX)) = E(eat ebtX) = eat Mx (bt) (i) [ ²* ³ x (2 − x) dx = ³ [x² - 2³²12] = ³ (4-8) = 1 (j) P(X > x) = P(0 events in [0, x]) = (Ax)⁰e-Ax 0! -λx so F(x) = 1- e-λx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer q's a, b, and c
![For each of the following proof excerpts, explain as fully as possible what the mathematical
statistics context is, and what steps are being taken (i.e. relate it to the content being assessed).
Standard notation for functions and parameters is used throughout. NB: you do not need to
complete the proof.
(a) Σxen (ag(x) + bh(x)) f(x) = a Σxen 9(x)ƒ(x) +bΣxenh(x)ƒ(x)
(b) M"(t) = ²X(X – t)−¹ = & λ(A − t)−² = 2\(\ — t)−³ and M″(0) = 2/1/2
dt2
(c) P(Z² < x) = P( - √x < Z < √√x) = Fz(√x) – Fz( − √x)
∞
d
(d) Σxpq* = pqΣxq-1 = pq
dq
x=0
(e) F(2.5) 1 - e
=
-0.35x2.5
∞
d
-1
Σ9²
Σ² = pq dq (1 — q)-¹
x=0
= 0.583 (3dp)
n
n
tx
n-x
(1) Σeta
* (*) p² (1 − p)¹-² = Σ (*) (pe²)² (1 − p)¹−² = (1 − p + pe²)"
n-x
x=0
x=0
(g) F(x)=√x for 0 < x < 1 and F-¹(x) = x², so X = U²
(h) E(et(a+bX)) = E(eat ebtX) = eªt Mx (bt)
(i) [*²*¾a(2 − x)dx = ³ [x² − }x³[6]
³ [2² - 2²³|²] = ³ (4- ) = 1
(Ax)⁰e-A
(j) P(X > x) = P(0 events in [0, x]) =
=
Xx
= e
so F(x) = 1 – e-λx
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff76b43b9-ccee-44f5-a918-27c1f81bc882%2F7018b37b-ee59-45a9-8612-5c8bc37ef44b%2Fwfuj3oq_processed.png&w=3840&q=75)
Transcribed Image Text:For each of the following proof excerpts, explain as fully as possible what the mathematical
statistics context is, and what steps are being taken (i.e. relate it to the content being assessed).
Standard notation for functions and parameters is used throughout. NB: you do not need to
complete the proof.
(a) Σxen (ag(x) + bh(x)) f(x) = a Σxen 9(x)ƒ(x) +bΣxenh(x)ƒ(x)
(b) M"(t) = ²X(X – t)−¹ = & λ(A − t)−² = 2\(\ — t)−³ and M″(0) = 2/1/2
dt2
(c) P(Z² < x) = P( - √x < Z < √√x) = Fz(√x) – Fz( − √x)
∞
d
(d) Σxpq* = pqΣxq-1 = pq
dq
x=0
(e) F(2.5) 1 - e
=
-0.35x2.5
∞
d
-1
Σ9²
Σ² = pq dq (1 — q)-¹
x=0
= 0.583 (3dp)
n
n
tx
n-x
(1) Σeta
* (*) p² (1 − p)¹-² = Σ (*) (pe²)² (1 − p)¹−² = (1 − p + pe²)"
n-x
x=0
x=0
(g) F(x)=√x for 0 < x < 1 and F-¹(x) = x², so X = U²
(h) E(et(a+bX)) = E(eat ebtX) = eªt Mx (bt)
(i) [*²*¾a(2 − x)dx = ³ [x² − }x³[6]
³ [2² - 2²³|²] = ³ (4- ) = 1
(Ax)⁰e-A
(j) P(X > x) = P(0 events in [0, x]) =
=
Xx
= e
so F(x) = 1 – e-λx
-
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