THEOREM 4.1 (The Linearity Property of Expectation). Let X and Y be two random variables. Then the expectation of their sum is the sum of their expectations; that is, if Z = X + Y, then E[Z] = E[X+Y] = E[X] + E[Y]. Proof: We will prove the theorem assuming that X, Y, and hence Z are continuous random variables. Proof for the discrete case is very similar. E[X + Y)= ( (x+y)f(x,y) dr dy = [² f(x,y) dy dx + f(x,y)dx dy I Y
THEOREM 4.1 (The Linearity Property of Expectation). Let X and Y be two random variables. Then the expectation of their sum is the sum of their expectations; that is, if Z = X + Y, then E[Z] = E[X+Y] = E[X] + E[Y]. Proof: We will prove the theorem assuming that X, Y, and hence Z are continuous random variables. Proof for the discrete case is very similar. E[X + Y)= ( (x+y)f(x,y) dr dy = [² f(x,y) dy dx + f(x,y)dx dy I Y
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.1: Angle Relationships
Problem 44E
Related questions
Question
![b
asX<Ysb
a<x< X<b
a
X
a
b
Figure 4.3. Two areas of integration for Example 4.7
2 [6³-a³
(b-a)² 6
² - 2/² (b² − a²) + ª²(b=²
a²(b-a)
a
3
Thus the expected seek distance is one third the maximum seek distance. Intuition
may have led us to the incorrect conclusion that the expected seek distance is half
of the maximum. (In practice, the expected seek distance is even smaller because of
correlations between successive requests [HUNT 1980, IBM 1997].)
Certain functions of random variables (e.g., sums), are of special interest
and are of considerable use.
THEOREM 4.1 (The Linearity Property of Expectation).
Let X and Y be two random variables. Then the expectation of
their sum is the sum of their expectations; that is, if Z = X + Y,
then E[Z] = E[X +Y] = E[X] + E[Y].
Proof: We will prove the theorem assuming that X, Y, and hence Z are
continuous random variables. Proof for the discrete case is very similar.
E[X + Y] =
**** (x + y) ƒ(x, y)dx dy
=
* x** f(x, y) dy dx +
I
+ [_y_f(x, y)dx dy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85743d3f-92dc-4950-b8a4-671e5ea73748%2Fd5dc91eb-8942-4018-8f98-159964ff6790%2Fz08mtb9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:b
asX<Ysb
a<x< X<b
a
X
a
b
Figure 4.3. Two areas of integration for Example 4.7
2 [6³-a³
(b-a)² 6
² - 2/² (b² − a²) + ª²(b=²
a²(b-a)
a
3
Thus the expected seek distance is one third the maximum seek distance. Intuition
may have led us to the incorrect conclusion that the expected seek distance is half
of the maximum. (In practice, the expected seek distance is even smaller because of
correlations between successive requests [HUNT 1980, IBM 1997].)
Certain functions of random variables (e.g., sums), are of special interest
and are of considerable use.
THEOREM 4.1 (The Linearity Property of Expectation).
Let X and Y be two random variables. Then the expectation of
their sum is the sum of their expectations; that is, if Z = X + Y,
then E[Z] = E[X +Y] = E[X] + E[Y].
Proof: We will prove the theorem assuming that X, Y, and hence Z are
continuous random variables. Proof for the discrete case is very similar.
E[X + Y] =
**** (x + y) ƒ(x, y)dx dy
=
* x** f(x, y) dy dx +
I
+ [_y_f(x, y)dx dy
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