Show that if X has density function f. then E [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) f ( x ) d x Hint: Using Theoretical Exercise 5.2, start with E [ g ( X ) ] = ∫ 0 ∞ P { g ( x ) > y } d y − ∫ 0 ∞ P { g ( x ) < − y } d y and then proceed as in the proof given in the text when g ( x ) > 0 .
Show that if X has density function f. then E [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) f ( x ) d x Hint: Using Theoretical Exercise 5.2, start with E [ g ( X ) ] = ∫ 0 ∞ P { g ( x ) > y } d y − ∫ 0 ∞ P { g ( x ) < − y } d y and then proceed as in the proof given in the text when g ( x ) > 0 .
Solution Summary: The author explains that E[g(X)] can be written as follows.
Show that if X has density function f. then
E
[
g
(
X
)
]
=
∫
−
∞
∞
g
(
x
)
f
(
x
)
d
x
Hint: Using Theoretical Exercise 5.2, start with
E
[
g
(
X
)
]
=
∫
0
∞
P
{
g
(
x
)
>
y
}
d
y
−
∫
0
∞
P
{
g
(
x
)
<
−
y
}
d
y
and then proceed as in the proof given in the text when
g
(
x
)
>
0
.
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.
Ⓒ Let f = [0, too) → 1h with f(0) = 1 and
VXE[0, too): f'(x) = f(x).
a)
Analyze the function
with respect to monotonicity.
g(x) = f(x)/ex
A function y = 5x2 + 10x is defined over an open
interval x = (1, 2). Atleast at one point in this
interval, dy/dx is exactly
Lagrange's Mean Value Theorem.
according to
Consider the function f(x) = ln(x2). Let L(x) be the local linearization of f aboutx bar = a.a) What is the general expression of L(x) ?b) Find the linearization L(x) about a = 1 , a = 2 and a = 3.c) Which one of the above linearizations will fit the best for x = 2.5?
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY