x and n are independent the end to end transmission

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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 ([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...
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