Unformatted text preview: C lecture 8 7 / 26 Transmission Time of a File over a Wireless Network
* Assumptions: Ri (i = 1, 2, . . . ), X , and N are independent
The Endtoend transmission time, denoted by T , is given by
N T = X ∑ Ri i =1 E [T ] =?
Divide and conquer again
n2 E [T ] = ∑ n=n1 M. Chen ([email protected]) n pN (n) ∑ E ENGG2430C lecture 8 i =1 X
Ri 8 / 26 Continuous Random Variable and PDF R.v.s with a continuous range of possible values are common
Speed of a vehicle on a highway
Throwing a dart onto a board, position of the hitting point (From the textbook) PDF  Probability Density Function
PDF of a r.v. X , denoted by fX (x ), is a nonnegative function,
satisfying
ˆ
P (X ∈ B ) =
fX (x ) dx , ∀B ⊆ R .
B
M. Chen ([email protected]) ENGG2430C lecture 8 9 / 26 Expectation and Variance Let X be a continuous r.v. with PDF fX (x ), then
ˆ∞
x fX (x ) dx
E [X ] =
∞
ˆ−∞
E [g (X )] =
g (x )fX (x ) dx
−∞
ˆ∞
2
Var(X ) = E [(X − E [X ]) ] =
(x − E [X ])2 fX (x ) dx
−∞
2 2 = E [X ] − E [X ] ≥ 0
The expectation (i.e., mean) still represents the center of gravity and
the average performance.
Variance still captures how the mass concentrates around the mean.
M. Chen ([email protected]) ENGG2430C lecture 8 10 / 26 Uniform Random Variable Let X be a continuous r.v. with a uniform PDF fX (x ) (From the textbook) fX (x ) = E [X ] = E [X 2 ] = Var(X ) = 1
, a≤x ≤b
b −a
a+b
2
ˆb 2
x
1 13
dx =
x
b −a
b −a 3
a b
a = a2 + ab + b 2
3 E [X 2 ] − E 2 [X ] = (b − a)2 /12 E [2X + 3] = a + b + 3; Var(2X + 3) = (b − a)2 /3.
M. Chen ([email protected]) ENGG2430C lecture 8 11 / 26 Why Not Use PMF for Continuous R.R.? Let X be a continuous r.v. with PDF fX (x ), then
fX (x ) ≥ 0 for all x in R
ˆ
P (X ∈ B ) fX (x ) dx , ∀B ⊆ R = B
ˆb...
View
Full Document
 Spring '14
 Probability theory, M. Chen, Minghua Chen, engg2430c lecture

Click to edit the document details