Xi = 1 if ith access is a hit, 0 if miss. 1st miss on kth access 1st access=hit k-1 access=hit kth access miss.

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Presentation transcript:

Xi = 1 if ith access is a hit, 0 if miss. 1st miss on kth access 1st access=hit k-1 access=hit kth access miss

Marginal pmfs

Derivation of (2.8) and (2.9)

Solution...

2.4 Expectation

2.4 Expectation

Example 2.22 (Poisson RV)

Special Care is Required Sometimes

If Y = g(X); i.e., Y(ω) = g(X(ω)), to compute E[Y] requires the pmf of Y, or does it...

Law of the Unconscious Statistician (LOTUS) And Leads to linearity and other properties...

Moments nth moment: E[X n], n = 1, 2, 3, . . . nth central moment: E[(X-m)n], m := E[X] Variance = 2nd central moment σ 2 := var(X) := E[(X-m)2]

Variance formula: Derivation:

Example 2.29 (Poisson RV)

Indicator Functions

The Markov Inequality For X ≥ 0 and a > 0, follows from Derivation . . .

The Chebyshev Inequality

The Chebyshev Inequality To derive this, take X = |Y| and r = 2 in (2.19):