Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution


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1 Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin Rundel Februry 1, 2012 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Unit Norml Distribution The unit norml distribution is specil cse of the norml distribution where µ = 0 nd σ = 1, Z N (0, 1. P(Z = z = φ(z = 1 e 1 2 z2 The re under the unit norml curve from to is given by P(z = 1 e t2 /2 dt = Φ( Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
2 The re under the unit norml curve from to is given by P(z = 1 e t2 /2 dt = 1 Φ(x The re under the unit norml curve from to b where b is given by P( z b = 1 b e t2 /2 dt = Φ(b Φ( b Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Evluting Φ The re under the unit norml curve outside of to b where b is given by P( z b = 1 1 b e t2 /2 dt = Φ( + (1 Φ(b = 1 (Φ(b Φ( The function Φ(x hs no simple closed form solution, sometimes written in terms of the error function erf Φ(x = 1 x e x2 /2 = 1 [ ( ] x 1 + erf 2 2 where erf(x = 2 π x 0 e t2 This doesn t seem to get us very fr, but we cn tke the Tylor expnsion of e t2 nd evlute the integrl of ech term to get series for erf(x b Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
3 Evluting Φ  In prctice Evluting Φ  Some prctice 362 APPENDIX C. DISTRIBUTION TABLES In prctice you will never hve to evlute Φ explicitly, we either look Φ(x up in tble or use computer to clculte it (pnorm in R. Working with your neighbor(s see if you cn work out the following probbilities for rndom vrible Z N(0, 1. negtive Z Second deciml plce of Z Z Sttistics 104 (Colin Rundel Lecture : More Binomil Distribution Februry 1, / Chpter For Z 3.50, the probbility is less thn or equl to Empiricl Rule P(z < 1 P(z > 2.22 P( 1.53 z 2.75 P(0.75 z 1.43 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Stndrdizing Norml Distributions Everything we just discussed pplies only to the unit norml distribution, but this doesn t come up very often in useful problems. P( 1 z 1 = Φ(1 Φ( 1 = P( 2 z 2 = Φ(2 Φ( 2 = Let X be normlly distributed rndom vrible with men µ nd vrince σ 2 then we define the rndom vrible Z such tht ( x µ Z = N(0, 1 σ In previous clsses you hve probbly referred to this s zscore. We will see why this works soon when we get to functions of rndom vribles. P( x b = b ( 1 exp 1 σ 2 (x µ 2 b µ µ σ 2 dx = Φ Φ σ σ P( 3 z 3 = Φ(3 Φ( 3 = Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
4 Another wy to see DeMoivreLplce We sw nother useful pproximtion lst week  Stirling s pproximtion to the fctoril function which is prticulrly good for lrge n. n! n n n e n Stirling s pproximtion is bsed on the Stirling Series n! = ( n ( n 1 n 1 + e 2 1 (6n (6n (2 3 5(6n (2 3 5(6n 4 + We strt with the definition of the Binomil distribution nd use Sterling s pproximtion for the fctorils n! P(X = k = (n k! k! pk (1 p n k n n n e n (n k(n k n k e n+k ( k k k e k pk q (n k n k(n k ( np ( k nq n k k n k Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Set x = k np, so k = np + x nd n k = nq x then ( 1 np np+x ( nq P(X = np + x npq np + x nq x ( np + x np x ( nq x c np nq nq+x c (1 + x/np np x (1 x/nq nq+x nq x Like lst time, we will use n pproximtion for log(1 ± x which is bsed on the Tylor expnsions log(1 + x = x 1 2 x x3 + ( 1 n 1 n xn log(1 x = x 1 2 x2 1 3 x3 1 n xn Tke the log of both sides of the eqution nd use the 1st nd 2nd term of the Tylor expnsion for the pproximtion to get log P(X = np + x c (np + x log(1 + x/np (nq x log(1 x/nq ( ( x c (np + x np x2 2(np 2 (nq x x nq x2 2(nq 2 P(X = k = c x x2 np + x2 2np + x3 2(np 2 + x x2 nq + x2 2nq c 2qx2 2npq + qx2 2npq + 2px2 2npq + px2 2npq + x2 c 2npq ( 1 exp 1 npq 2 (k np 2 npq x3 2(nq 2 ( x 3 2(np 2 x3 2(nq 2 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
5 Improving the pproximtion Improving the pproximtion, cont. Tke for exmple Binomil distribution where n = 20 nd p = 10, we should be ble to pproximte the distribution with X N (10, 5. Binomil probbility: 13 ( 20 P(7 x 13 = 0.5 x ( x x x=7 Nive pproximtion: P(7 x 13 Φ Φ It is cler tht our pproximtion is missing 1/2 of P(X = 7 nd P(X = 13, s n this error is very smll. In this cse P(X = 7 = P(X = 13 = so our pproximtion is off by 7%. Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Continuity corrected pproximtion: / /2 10 P(7 x 13 Φ Φ 5 5 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Improving the pproximtion, cont. Improving the pproximtion, cont. When we scle up n to 200 nd our rnge interest to P(70 x 130: Binomil probbility: Nive pproximtion: P(70 x 130 = P(70 x 130 Φ 130 x=70 ( 200 x 0.5 x ( x Φ Continuity corrected pproximtion: / /2 100 P(70 x 130 Φ Φ This correction lso lets us do, modertely useless, things like clculte the probbility for prticulr vlue of k. Such s, wht is the chnce of 50 Heds in 100 tosses of slightly unfir coin (p = 0.55? Binomil probbility: P(x = 50 = ( ( Nive pproximtion: P(x = 50 Φ Φ Continuity corrected pproximtion: / /2 55 P(x = 50 Φ Φ Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
6 Smple Problem (1 Smple Problem ( Roll fir die 500 times, wht s the probbility of rolling t lest 100 ones? A pollster wishes to know the percentge p of people in popultion who intend to vote for prticulr cndidte. How lrge must rndom smple with replcement be in order to be t lest 95% sure tht the smple percentge is within one percentge point of p? Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21 Sttistics 104 (Colin Rundel Lecture 6: More Binomil Distribution Februry 1, / 21
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