Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose value ca be calculated from the sample data. It s a radom varable wth a dstrbuto fucto. The radom varables, 2,, are sad to form a (smple) radom sample of sze f the s are depedet radom varables ad each has the sample probablty dstrbuto. We say that the s are d. week
Example Toss a co tmes. Suppose 0 f f th th toss came up H toss came up T s are Beroull radom varables wth p ½ ad E( ) ½. The proporto of heads s. It s a statstc. week 2
Samplg Dstrbuto of a Statstc The samplg dstrbuto of a statstc s the dstrbuto of values take by the statstc all possble samples of the same sze from the same populato. The dstrbuto fucto of a statstc s NOT the same as the dstrbuto of the orgal populato that geerated the orgal sample. Probablty rules ca be used to obta the dstrbuto of a statstc provded that t s a smple fucto of the s ad ether there are relatvely few dfferet values he populato or else the populato dstrbuto has a ce form. Alteratvely, we ca perform a smulato expermet to obta formato about the samplg dstrbuto of a statstc. week 3
Markov s Iequalty If s a o-egatve radom varable wth E() < ad a >0 the, Proof: P ( a) E( ) a week 4
Chebyshev s Iequalty For a radom varable wth E() < ad V() <, for ay a >0 Proof: P ( E( ) a) V a ( ) 2 week 5
week 6 Law of Large Numbers Iterested sequece of radom varables, 2, 3, such that the radom varables are depedet ad detcally dstrbuted (..d). Let Suppose E( ) μ, V( ) σ 2, the ad Itutvely, as, so ( ) ( ) μ E E E ( ) ( ) V V V 2 2 σ ( ) 0 V ( ) μ E
Formally, the Weak Law of Large Numbers (WLLN) states the followg: Suppose, 2, 3, are..d wth E( ) μ <, V( ) σ 2 <, the for ay postve umber a as. Ths s called Covergece Probablty. Proof: ( a) 0 P μ week 7
Flp a co 0,000 tmes. Let 0 E( ) ½ ad V( ) ¼. Example toss came up T Take a 0.0, the by Chebyshev s Iequalty P 2 Chebyshev Iequalty gves a very weak upper boud. f f th 0.0 th toss came up H 4 0,000 ( ) ( ) 4 0.0 Chebyshev Iequalty works regardless of the dstrbuto of the s. The WLLN state that the proportos of heads the 0,000 tosses coverge probablty to 0.5. 2 week 8
Strog Law of Large Number Suppose, 2, 3, are..d wth E( ) μ <, the coverges to μ as wth probablty. That s P lm 2 ( + + L + ) μ Ths s called covergece almost surely. week 9
Cetral Lmt Theorem The cetral lmt theorem s cocered wth the lmtg property of sums of radom varables. If, 2, s a sequece of..d radom varables wth mea μ ad varace σ 2 ad, S the by the WLLN we have that μ probablty. The CLT cocered ot just wth the fact of covergece but how S / fluctuates aroud μ. Note that E(S ) μ ad V(S ) σ 2. The stadardzed verso of S s S Z S μ σ ad we have that E(Z ) 0, V(Z ). week 0
The Cetral Lmt Theorem Let, 2, be a sequece of..d radom varables wth E( ) μ < ad Var( ) σ 2 <. Let S The, S μ lm P σ z P ( Z z) Φ( z) for - < x < where Z s a stadard ormal radom varable ad Ф(z)s the cdf for the stadard ormal dstrbuto. S μ Ths s equvalet to sayg that Z coverges dstrbuto to σ Z ~ N(0,). Also, lm P σ μ x Φ ( x) μ.e. Z coverges dstrbuto to Z ~ N(0,). σ week
Example Suppose, 2, are..d radom varables ad each has the Posso(3) dstrbuto. So E( ) V( ) 3. ( ) ( ) The CLT says that P + + 3 + x 3 Φ x as. L week 2
Examples A very commo applcato of the CLT s the Normal approxmato to the Bomal dstrbuto. Suppose, 2, are..d radom varables ad each has the Beroull(p) dstrbuto. So E( ) p ad V( ) p(-p). ( ( )) ( ) The CLT says that P + + p x p p x as. + Φ L Let Y + + the Y has a Bomal(, p) dstrbuto. So for large, P ( Y y) P Y p p y p Φ y p ( p) p( p) p( p) Suppose we flp a based co 000 tmes ad the probablty of heads o ay oe toss s 0.6. Fd the probablty of gettg at least 550 heads. Suppose we toss a co 00 tmes ad observed 60 heads. Is the co far? week 3
Samplg from Normal Populato If the orgal populato has a ormal dstrbuto, the sample mea s also ormally dstrbuted. We do t eed the CLT ths case. I geeral, f, 2,,..d N(μ, σ 2 ) the S + 2 + + ~ N(μ, σ 2 ) ad S ~ 2 σ N μ, week 4
Example week 5