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Spherical Error Probable (SEP) speci es the radius of the sphere centered on correct location that is expected to contain 50% of the realizations variable w In the special case where = 1 = 2 = 3 , then eqn (4129) simpli es to the equation for the sphere w = x 154 In all other cases, the probability ellipsoid expected to contain 50% of the realizations is de ned by eqn (4129) with c = 154 Various approximations exist for estimating the radius of the SEP sphere [39] 4912 Analysis in Two Dimensions

104 k

When you come across a sum or difference that is multiplied by a single number or variable, you will sometimes want to expand it into a sum or difference of products To do this, you can use the distributive laws

The analysis of this section is similar to that of Section 4911, but performed in two dimensions The two dimensions that are of interest are typically the horizontal (eg, north and east) components In this section, we will use the vectors w, v, x which have interpretations parallel to w, v, x from Section 4911 The vector symbol over the symbols indicates that the corresponding variable is an element of R2 The joint horizontal position error density in the w coordinates is pw = 2 1 |P| 1 exp w P 1 w 2 (4130)

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Multiplication over addition Suppose you have three integers a, b, and c arranged so that you must multiply a by the sum of b and c The left-hand distributive law of multiplication over addition tells you that this is equal to the product ab plus the product bc This comes out simpler if you write it down:

a(b + c) = ab + ac Now imagine multiplying the sum of a and b by c The right-hand distributive law of multiplication over addition says that (a + b)c = ac + bc

The transformation from the three dimensional distribution of eqn (4128) to the two dimensional distribution of eqn (4130) is discussed in Exercise 410 The joint horizontal position error density in the x coordinates is px = 1 exp 2 1 2 x2 x2 1 + 22 2 2 1 2 2 (4131)

Value 15 pF 33 pF 10 pF 15 pF 20 pF 30 pF 33 pF 47 pF 56 pF 68 pF 75 pF 82 pF 91 pF 100 pF 120 pF 130 pF 150 pF 180 pF 220 pF 330 pF 470 pF 560 pF 680 pF 750 pF 820 pF

Multiplication over subtraction The distributive laws also work with subtraction For any three integers a, b, and c

(4132)

a(b c) = ab ac and (a b)c = ac bc It s not hard to show how these follow from the laws for addition If you want to do a rigorous job, the process is rather long Table 5-3 breaks the derivation down into an S/R process, showing every logical step, for the left-hand distributive law of multiplication over subtraction If you want to do a proof for the right-hand distributive law of multiplication over subtraction, consider it a bonus exercise!

Type Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic

Circular Error Probable (CEP) is the radius of the circle centered on the correct location that contains 50% of the expected horizontal position errors The R95 statistic is the radius of the circle expected to contain 95% of the horizontal position errors The analysis to de ne both of these statistics is the same Therefore, we will perform the analysis for probability [0, 1] At the conclusion of the analysis, we can compute the results for = 05 or 095 Let = P rob{ v c},

The left-hand law fails with division The left-hand distributive laws do not work for division over addition or for division over subtraction To see why, all you have to do is produce examples of failure That s easy! Consider this:

24/(4 + 2) = 24/6 = 4 but 24/4 + 24/2 = 6 + 12 = 18 And this: 24/(4 2) = 24/2 = 12

then for S, U, , de ned as in the previous section, = P rob{ v c} = P rob{w P 1 w c2 } = P rob{x

Table 5-3 Derivation of the left-hand distributive law for multiplication over subtraction As you read down, each statement is equal to all the statements above it Warning: Don t mistake the expression ( 1) for the subtraction of 1! The parentheses emphasize that 1 is a factor in a product

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