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 Σx≤n 9(x)f(x) +bΣx≤Nh(x)f(x) (b) M"(t) = ¹⁄² \(\ − t)−¹ = &λ(A − t)−² = 2\(\ — t)−³ and M″(0) = 31/12 (c) P(Z² < x) = P( − √x < Z < √x) = Fz(√x) – Fz( − √x) ∞ ∞ d x-1 (d) Σxpq* = = pq Σxq²- = paw Σq* = pq = (1 – g)-1 dq dq x=0 x=1 x=0 (e) F(2.5) = 1 – e-0.35×2.5 = 0.583 (3dp) n n (1) Σ e¹² (²) p² (1 − p)¹−² = Σ (") (pe¹)² (1 − p)"−² = (1 − p + pe¹)" etx 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) [ ²³ ³x(2 − x)dx = ³ [x² − }\x³|6] = {( 4 − }) = 1 0 (j) P(X > x) = P(0 events in [0, x]) = (λx)⁰e¬^ 0! 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) = a Σx≤n 9(x)f(x) +bΣx≤Nh(x)f(x) (b) M"(t) = ¹⁄² \(\ − t)−¹ = &λ(A − t)−² = 2\(\ — t)−³ and M″(0) = 31/12 (c) P(Z² < x) = P( − √x < Z < √x) = Fz(√x) – Fz( − √x) ∞ ∞ d x-1 (d) Σxpq* = = pq Σxq²- = paw Σq* = pq = (1 – g)-1 dq dq x=0 x=1 x=0 (e) F(2.5) = 1 – e-0.35×2.5 = 0.583 (3dp) n n (1) Σ e¹² (²) p² (1 − p)¹−² = Σ (") (pe¹)² (1 − p)"−² = (1 − p + pe¹)" etx 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) [ ²³ ³x(2 − x)dx = ³ [x² − }\x³|6] = {( 4 − }) = 1 0 (j) P(X > x) = P(0 events in [0, x]) = (λx)⁰e¬^ 0! so F(x) = 1- e-λx
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please answer q j
![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 = pq7q²
dq
x=0
(e) F(2.5) 1 - e
=
∞
d
-1
Σ² = pq dq (1 — q)−¹
x=0
-0.35x2.5
= 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]) =
=
= 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%2F60edc63a-8dd4-45df-bc64-e785fc389df8%2F98d1i_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 = pq7q²
dq
x=0
(e) F(2.5) 1 - e
=
∞
d
-1
Σ² = pq dq (1 — q)−¹
x=0
-0.35x2.5
= 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]) =
=
= e
so F(x) = 1 - e¯λx
Expert Solution
Step 1
j) For the given proof, the context is related to the derivation of the cumulative distribution function (CDF) of the exponential distribution using the Poisson distribution.
In other words, the length of the time interval between the successive occurrences of an event has an exponential distribution provided that the number of occurrences in a fixed time interval follows the Poisson distribution.
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