NotesOct15

# NotesOct15 - A linear combination of two independent normal...

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A linear combination of two independent normal random variables Let X 1 N( μ 1 , σ 2 1 ) and X 2 N( μ 2 , σ 2 2 ) be independent normal random variables. Let a and b be two numbers. Let Y = aX 1 + bX 2 . Then Y N( 1 + 2 , a 2 σ 2 1 + b 2 σ 2 2 ) .

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Linear combinations of several independent normal random variables Let X i N ( μ i , σ 2 i ) , i = 1 , . . . , n, be inde- pendent normal random variables. Let a 1 , . . . , a n be numbers. Let Y = n i =1 a i X i . Then Y N( n X i =1 a i μ i , n X i =1 a 2 i σ 2 i ) .
Normal approximations 1 Recall the normal approximation to the bi- nomial distribution . If X Bin( n, p ), n is large and p is not too close to either 0 or 1, then Y = X - np q np (1 - p ) has, approximately, the standard normal distri- bution N(0 , 1). Rule of thumb : np > 5 and n (1 - p ) > 5.

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2 The normal approximation to the Poisson distribution . If X Poiss( λ ) and λ is large, then Y = X - λ λ has, approximately, the standard normal distri- bution N(0 , 1).
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