Probability Distributions. Statistics and Quantitative Analysis U4320. Probability Distributions(cont.) Probability

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1 Statstcs and Quanttatve Analss U430 Dstrbutons A. Dstrbutons: How do smple probablt tables relate to dstrbutons?. What s the of gettng a head? ( con toss) Prob. Segment 4: Dstrbutons, Unvarate & Bvarate / Prof. Sharn O Halloran 0 Proporton of Heads. ow sa we flp the con twce. The pcture now looks lke: As number of con tosses ncreases, the dstrbuton looks lke a bell-shaped curve: / /4 0 / Proporton of Heads 0 head s head heads 0 / Proporton of Heads

2 3. General: ormal Dstrbuton dstrbutons are dealzed bar graphs or hstograms. As we get more and more tosses, the probablt of an one observaton falls to zero. Thus, the fnal result s a bell-shaped curve B. Propertes of a ormal Dstrbuton. Formulas: Mean and Varance Mean Varance Populaton X µ = = ( X µ ) σ = = X s Sample n X = = n n ( X X ) = = n. ote Dfference wth the Book The populaton mean s wrtten as: µ p( ), Varance as:. ote Dfference wth the Book (cont.) So the average, or epected, number of heads n two tosses of a con s: 0*/4 + */ + */4 =. σ ( µ ) p( ). Eample: Two tosses of a con umber of Heads p() 0 /4 / /4

3 3. Epected Value E() = Average or Mean [ ( ) ] E µ = Epected Varance C. Standard ormal Dstrbuton.Defnton: a normal dstrbuton wth mean 0 and standard devaton. Total Area of Curve = - Are ponts on the -as that show how man standard devatons σ= that pont s awa from the mean m. C. Standard ormal Dstrbuton. Characterstcs smmetrc Unmodal. contnuous dstrbutons 3. Eample: Heght of people are normall dstrbuted wth mean 5'7" D. How to Calculate Z-scores Defnton: Z-value s the number of standard devatons awa from the mean Defnton: Z-tables gve the probablt (score) of observng a partcular z-value. Total Area of Curve = area=/ What s the proporton of people taller than 5'7"? 5'7" =µ 3

4 . What s the area under the curve that s greater than? Prob (Z>) The entr n the table s 0.59, whch s the total area to the rght of.. What s the area to the rght of.64? Prob (Z >.64) The table gves 0.05, or about 5%. Total Area of Curve = Total Area of Curve = z= What s the area to the left of -.64? Prob (Z < -.64) 4. What s the probablt that an observaton les between 0 and? Prob (0 < Z < ) Total Area of Curve =

5 5. How would ou fgure out the area between and.5 on the graph? Prob ( < Z <.5) What s the area between - and? Prob (- < Z < ) P (-<Z<0) =.34 P (0<z<) = = = What s the area between - and? Prob(-<Z<) - Prob (Z< -) - Prob (Z>) = =.954 E. Standardzaton. Standard ormal Dstrbuton -- s a ver specal case where the mean of dstrbuton equals 0 and the standard devaton equals σ=.00 σ=.00 5

6 . Case : Standard devaton dffers from For a normal dstrbuton wth mean 0 and some standard devaton, ou can convert an pont to the standard normal dstrbuton b changng t to /. SD= 3. Case : Mean dffers from 0 So startng wth an normal dstrbuton wth mean and standard devaton, ou can convert to a standard normal b takng - and usng ths as our Z-value. ow, what would be the area under the graph between 50 and 5? Prob (0<Z<) =.34 SD=3 SD= -.00' ' 5 =5 4. General Case: Mean not equal to 0 and SD not equal to Sa ou have a normal dstrbuton wth mean & standard devaton. You can convert an pont n that dstrbuton to the same pont n the standard normal b computng Z = µ σ. Ths s called standardzaton. The Z-value corresponds to. The Z-table lets ou look up the Z-Score of an number. 5. Trout Eample: a. The lengths of trout caught n a lake are normall dstrbuted wth mean 9.5" and standard devaton.4". There s a law that ou can't keep an fsh below ". What percent of the trout s ths? (Can keep above ) Step : Standardze Fnd the Z-score of : Prob (>) Z = = Step : Fnd z-score Fnd Prob (Z>.79) Look up.79 n our table; onl.037, or about 4% of the fsh could be kept. 6

7 5. Trout Eample (cont.): b. ow the're thnkng of changng the standard to 0" nstead of ". What proporton of fsh could be kept under the new lmt? Standardze Prob (>0) Z = = Step : Fnd z-score Prob (Z>.36) In our tables, ths gves.359, or almost 36% of the fsh could be kept under the new law. Jont Dstrbutons A. Tables. Eample: Toss a con 3 tmes. How man heads and how man runs do we observe? Def: A run s a sequence of one or more of the same event n a row Possble outcomes Toss Heads Runs TTT /8 0 TTH /8 THT /8 3 THH /8 HTT /8 HTH /8 3 HHT /8 HHH /8 3 Jont Dstrbutons (cont.). Jont Dstrbuton Table Runs Heads 3 0 /8 0 0 /8 0 /4 (/8) /8 3/8 0 /4 (/8) /8 3/8 3 /8 0 0 /8 /4 / /4 Jont Dstrbutons (cont.) 3. Defnton: The jont probablt of and s the probablt that both and occur. p(,) = Pr(X and Y) p(0, ) = /8, p(, ) = /4, and p(3, 3) = 0. 7

8 Jont Dstrbutons (cont.) B. Margnal Probabltes. Def: Margnal probablt s the sum of the rows and columns. The overall probablt of an event occurrng. p( ) = p(, ). So the probablt that there are just head s the prob of head and runs + head and runs + head and 3 runs = 0 + /4 + /8 = 3/8 Jont Dstrbutons (cont.) C. Independence A and B are ndependent f P(A B) = P(A). P( A& B) P( A B) = ; P( B) P( A) = P( A B), P( A, B) = P ( A) P ( B). Jont Dstrbutons (cont.) Are the # of heads and the # of runs ndependent? # Runs # heads 3 marg dst 0 /8 3/8 3/8 3 /8 marg dst /4 / /4 Correlaton and A..Defnton of Whch s defned as the epected value of the product of the dfferences from the means. σ, = E ( X µ )( Y µ Y ) = = ( X µ )( Y µ ) Y = ( X µ )( Y µ ) p(, ). Y 8

9 Correlaton and.graph Correlaton and B. Correlaton.Defnton of Correlaton ρ σ, = = σ σ SD * SD Correlaton and. Characterstcs of Correlaton - ρ f ρ = then Correlaton and. Characterstcs of Correlaton (cont.) f ρ = - 9

10 Correlaton and. Characterstcs of Correlaton (cont.) Correlaton and. Characterstcs of Correlaton (cont.) Wh? f ρ = 0 σ, ρ = = σ σ n ( µ )( µ ) ( µ ) ( µ ) = = n 0