00095 ¸ef4b21fd40ed4259ad7865fe8cfe4c

94

Hurwitz & Mathur

= (W(l + PO) x (1 + p,)/(l + P2)

= L,/(l+p₂)    [A6]

By equation [Al]

Lo = 1 + L0P0+ Li pi + L2P2

Substituting values of Li and L2 from [A5] and [A6]

Lo=l+LoPo + Lo ^xpi /(I +p,)+ Lo^Pz/Cl+Pz)

1 =Lo-LoPo -U x p,/(I+p,) -Lo^xp2/(l+p2)

Extracting L₀ as a common factor from the RHS, and notÅ‚ng that the ARL = L by eÄ…uation [A4], we obtain [1], namely

ARL = |1-Po-P,/(1 + Pi)-P2/(1+P2)]''

EXAMPLE: for m = 1.0 (s = 1.0) in Table 1

From the standardized (z) Normal distribution tables,

Po = P(-1.5 < x < 1.5) = P(-2.5 < z < 0.5) = 0.6853 p, = P( 1.5 < x < 3.0) = P( 0.5 < z < 2.0) = 0.2857 P2 = P(-3.0 < x <-1.5) = P(-4.0 < z <-2.5) = 0.0062

Substituting po, pi and P2 values in the above formula, obtain ARL=11.58