Parameter, Statistic and Random Samples

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1 Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e., it is a quatity whose value ca be calculated from the sample data. It is a radom variable with a distributio fuctio. Statistics are used to make iferece about ukow populatio parameters. The radom variables X, X,, X are said to form a (simple) radom sample of size if the X i s are idepedet radom variables ad each X i has the sample probability distributio. We say that the X i s are iid. week

2 Example Sample Mea ad Variace Suppose X, X,, X is a radom sample of size from a populatio with mea μ ad variace σ. The sample mea is defied as X i X i. The sample variace is defied as S ( X i X ). i week

3 Goals of Statistics Estimate ukow parameters μ ad σ. Measure errors of these estimates. Test whether sample gives evidece that parameters are (or are ot) equal to a certai value. week 3

4 Samplig Distributio of a Statistic The samplig distributio of a statistic is the distributio of values take by the statistic i all possible samples of the same size from the same populatio. The distributio fuctio of a statistic is NOT the same as the distributio of the origial populatio that geerated the origial sample. The form of the theoretical samplig distributio of a statistic will deped upo the distributio of the observable radom variables i the sample. week 4

5 Samplig from Normal populatio Ofte we assume the radom sample X, X, X is from a ormal populatio with ukow mea μ ad variace σ. Suppose we are iterested i estimatig μ ad testig whether it is equal to a certai value. For this we eed to kow the probability distributio of the estimator of μ. week 5

6 Claim Suppose X, X, X are i.i.d ormal radom variables with ukow mea μ ad variace σ the X ~ σ N μ, Proof: week 6

7 Recall - The Chi Square distributio If Z ~ N(0,) the, X Z has a Chi-Square distributio with parameter, i.e., X χ ~ (). Ca proof this usig chage of variable theorem for uivariate radom variables. The momet geeratig fuctio of X is m X () t t / If X χ, X ~ χ, K, X χ, all idepedet the Proof ~ k ( v ) ( v ) k ( v ) ~ k T ~ χ i X i Σ k v i week 7

8 Claim Suppose X, X, X are i.i.d ormal radom variables with mea μ ad variace σ. The, i Z are idepedet stadard ormal i σ variables, where i,,, ad Proof: i Z i X i μ X i μ σ ~ χ ( ) week 8

9 t distributio Suppose Z ~ N(0,) idepedet of X ~ χ (). The, T Z X / v ~ t ( ). v Proof: week 9

10 Claim Suppose X, X, X are i.i.d ormal radom variables with mea μ ad variace σ. The, Proof: X μ ~ t S / ( ) week 0

11 F distributio Suppose X ~ χ () idepedet of Y ~ χ (m). The, X / Y / m ~ F (, m) week

12 Properties of the F distributio The F-distributio is a right skewed distributio. F( ) i.e. m, F ( < a) P F(, m) (, m) P F (, m) > a P F( m, ) > a Ca use Table 7 o page 796 to fid percetile of the F- distributio. Example week

13 The Cetral Limit Theorem Let X, X, be a sequece of i.i.d radom variables with E(X i ) μ < ad Var(X i ) σ <. Let S X i i The, S μ lim P σ z P ( Z z) Φ( z) for - < x < where Z is a stadard ormal radom variable ad Ф(z)is the cdf for the stadard ormal distributio. S μ This is equivalet to sayig that Z coverges i distributio to σ Z ~ N(0,). X Also, lim P σ μ x Φ ( x) X μ i.e. Z coverges i distributio to Z ~ N(0,). σ week 3

14 Example Suppose X, X, are i.i.d radom variables ad each has the Poisso(3) distributio. So E(X i ) V(X i ) 3. ( ) ( ) The CLT says that P X + + X 3 + x 3 Φ x as. L week 4

15 Examples A very commo applicatio of the CLT is the Normal approximatio to the Biomial distributio. Suppose X, X, are i.i.d radom variables ad each has the Beroulli(p) distributio. So E(X i ) p ad V(X i ) p(-p). ( ( )) ( ) The CLT says that P X + + X p x p p x as. + Φ L Let Y X + + X the Y has a Biomial(, p) distributio. So for large, P ( Y y) P Y p p y p Φ y p ( p) p( p) p( p) Suppose we flip a biased coi 000 times ad the probability of heads o ay oe toss is 0.6. Fid the probability of gettig at least 550 heads. Suppose we toss a coi 00 times ad observed 60 heads. Is the coi fair? week 5

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