Let X 1 , . . .. X n , be independent and identically distributed random variables having distribution function F and density f. The quantity M ≡ [ X ( 1 ) + X ( n ) ] 2 defined to be the average of the smallest and largest values in X 1 , . . .. X n , is called the midrange of the sequence. Show that its distribution function is F M ( m ) = n ∫ − ∞ m [ F ( 2 m − x ) − F ( x ) ] n − 1 f ( x ) d x
Let X 1 , . . .. X n , be independent and identically distributed random variables having distribution function F and density f. The quantity M ≡ [ X ( 1 ) + X ( n ) ] 2 defined to be the average of the smallest and largest values in X 1 , . . .. X n , is called the midrange of the sequence. Show that its distribution function is F M ( m ) = n ∫ − ∞ m [ F ( 2 m − x ) − F ( x ) ] n − 1 f ( x ) d x
Solution Summary: The author explains the distribution function of given equation. The quantity M is the average of the smallest and largest values in X1,mathrm.....
Let
X
1
,
.
.
..
X
n
,
be independent and identically distributed random variables having distribution function F and density f. The quantity
M
≡
[
X
(
1
)
+
X
(
n
)
]
2
defined to be the average of the smallest and largest values in
X
1
,
.
.
..
X
n
,
is called the midrange of the sequence. Show that its distribution function is
F
M
(
m
)
=
n
∫
−
∞
m
[
F
(
2
m
−
x
)
−
F
(
x
)
]
n
−
1
f
(
x
)
d
x
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
Q prove or disprove: If Ely/x) = x = c(dipy
=BCCo
(BVC)
ECxly)=y, and E(X2), Ely)
In a small office, there are m = 5 typists who need to use a single typewriter to complete their reports. Assume the time
each typist takes to prepare a report follows an exponential distribution with an average of 20 minutes per preparation
(A = 3 reports/hour), and the service time for the typewriter to type out a report also follows an exponential distribution,
averaging 30 minutes to complete a report (μ 2 reports/hour). Given that the number of typists is finite and all typists
=
share one typewriter, they will form a waiting queue.
(1). Describe this queuing system and explain how it fits the characteristics of the M/M/1/∞0/m model.
(2). Calculate the probability that any typist is using the typewriter at steady-state.
(3). Calculate the average number of typists waiting in the queue at steady-state.
(4). Considering the need to reduce waiting time, if an additional typewriter is introduced (turning into a two-server
system, or M/M/2/∞0/m model), analyze the expected impact,…
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning