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.
ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rule
Let X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .
Let P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find it
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