summation - 1 The Summation Operator Because many...

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Unformatted text preview: 1 The Summation Operator Because many elementary propositions in econometrics involve the use of sums of numbers, it should be useful to review the summation operator, i.e. Σ. Assume a random variable (henceforth r.v) denoted X from which a sample of n quantities are observed, i.e., X i ,i = 1 ,...,n . Then the total sum of the observations ( X 1 + X 2 + ··· + X n ) can be represented as n X i =1 X i = X 1 + X 2 + ··· + X n (1) The following summation operator rules are useful. Learn them well. Rule 1. The summation of a constant k times a r.v X i is equal to the constant times the summation of that r.v. n X i =1 kX i = k n X i =1 X i (2) PROOF) n X i =1 kX i = kX 1 + kX 2 + ··· + kX n = k ( X 1 + X 2 + ··· + X n ) = k n X i =1 X i Rule 2. The summation of the sum of observations on two r.v’s is equal to the sum of their summations. n X i =1 ( X i + Y i ) = n X i =1 X i + n X i =1 Y i (3) PROOF) n X i =1 ( X i + Y i ) = ( X 1 + Y 1 ) + ( X 2 + Y 2 ) + ··· + ( X n + Y n...
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summation - 1 The Summation Operator Because many...

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