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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 End-to-end 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 (IE@CUHK) 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,
P (X ∈ B ) =
fX (x ) dx , ∀B ⊆ R .
M. Chen (IE@CUHK) ENGG2430C lecture 8 9 / 26 Expectation and Variance Let X be a continuous r.v. with PDF fX (x ), then
x fX (x ) dx
E [X ] =
E [g (X )] =
g (x )fX (x ) dx
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 (IE@CUHK) 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 3
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 (IE@CUHK) 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
P (X ∈ B ) fX (x ) dx , ∀B ⊆ R = B
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- Spring '14