{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 100AHW6 - STAT 100A HWVI Solution Problem 1 Suppose we flip...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT 100A HWVI Solution Problem 1: Suppose we flip a fair coin n times independently. Let X be the number of heads. Let k = n/ 2 + z √ n/ 2, or z = ( k- n/ 2) / ( √ n/ 2). Let g ( z ) = P ( X = k ). (1) Using the Stirling formula n ! ∼ √ 2 πnn n e- n , show that g (0) ∼ 1 √ 2 π 2 √ n . a ∼ b means that a/b → 1 as n → ∞ . g (0) = P ( X = n/ 2) = ˆ n n/ 2 ! / 2 n = n ! ( n/ 2)!( n/ 2)!2 n = √ 2 πnn n e- n [ p 2 π ( n/ 2)( n/ 2) n/ 2 e- n/ 2 ] 2 2 n = 1 √ 2 π 2 √ n . (2) Show that g ( z ) /g (0) → e- z 2 / 2 as n → ∞ . A: Let d = z √ n/ 2, g ( z ) g (0) = ( n n/ 2+ d ) ( n n/ 2 ) = n ! / [( n/ 2 + d )!( n/ 2- d )!] n ! / [( n/ 2)!( n/ 2)!] = ( n/ 2)!( n/ 2)! ( n/ 2 + d )!( n/ 2- d )! = ( n/ 2)( n/ 2- 1) ... ( n/ 2- d + 1) ( n/ 2 + 1) ... ( n/ 2 + d ) = (1- δ ) ... (1- ( d- 1) δ ) (1 + δ ) ... (1 + dδ ) ≈ e- ( δ + ... +( d- 1) δ ) e δ + ... + dδ = e- d ( d- 1) δ/ 2- d ( d +1) δ/ 2 = e- d 2 δ/ 2 = e- z 2 / 2 , where δ = 2 /n , and the “ ≈ ” becomes “=” as n → ∞ . So g ( z ) ∼ 1 √ 2 π e- z 2 / 2 2 √ n . (3) For two integers a < b , let a = ( a- n/ 2) / ( √ n/ 2), and b = ( b- n/ 2) / ( √ n/ 2). Show that P ( a ≤ X ≤ b ) → R b a f ( z ) dz , where f ( z ) = 1 √ 2 π e- z 2 / 2 . A: P ( a ≤ X ≤ b ) = b X k = a P ( X = k ) = b X z = a g ( z ) ≈ b X z = a 1 √ 2 π e- z 2 / 2 2 √ n = b X z = a f ( z )Δ z → Z b a f ( z ) dz, where Δ z = 2 / √ n , which is the space between every two consecutive values of z . “ ≈ ” becomes “=” as n → ∞ . (4) Let Z = ( X- n/ 2) / ( √ n/ 2). Show that P ( a ≤ X ≤ b ) = P ( a ≤ Z ≤ b ). Argue that in the limit Z ∼ N(0 , 1)....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

100AHW6 - STAT 100A HWVI Solution Problem 1 Suppose we flip...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online