Random Variables. Discrete Random Variables. Example of a random variable. We will look at: Nitrous Oxide Example. Nitrous Oxide Example

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1 Radom Varables Dscrete Radom Varables Dr. Tom Ilveto BUAD 8 Radom Varables varables that assume umercal values assocated wth radom outcomes from a expermet Radom varables ca be: Dscrete Cotuous We wll look at: We wll look at the probablty Dstrbuto for Radom Varables Expected Values (mea) ad Varace of Radom Varables Specal Dstrbutos Bomal Dstrbuto Normal Dstrbuto Example of a radom varable Suppose we were recordg the umber of detsts that use trous oxde (laughg gas) ther practce We kow that 6% of detsts use the gas (a pror for ths problem, but most lkely based o a survey result). P(Yes) =.6 P(No) =. Let = umber of detsts a radom sample of fve detsts that use use laughg gas. We ca lst the values of our radom varable s a radom varable that ca take o the followg values:,,,,,

2 Ad the assg probabltes to each value of the radom varable Probablty Dstrbuto The last table shows the probablty dstrbuto for the dscrete radom varable The table s also referred as probablty dstrbuto table P() Propertes of probablty dstrbuto Each probablty s betwee ad The sum of the probabltes for all values of x s equal to How I assged probabltes: P(Yes) =.6 P(No) =. P() = (.)(.)(.)(.)(.) =. =. P() = (.6)(.)(.)(.)(.) x =.768 s the the umber of combatos wth oly doctor usg the gas: Yes No No No No Or use the C,r formula No Yes No No No! No No Yes No No = = = No No No Yes No!( )! No No No No Yes Note: we are assumg depedece P() What s the probablty of of detsts selected radomly Usg laughg gas?.778 What s the probablty of less tha of usg laughg gas? If I radomly selected detsts, how may would I expect to use laughg gas? If I radomly selected detsts, how may would I expect to use laughg gas? Expectato = We do t have a way to solve ths yet, but we wll

3 : Graph of the Probablty dstrbuto Probablty Dstrbuto of p().... Number of Detsts Types of Radom Varables The varable the detst example s called a dscrete radom varable Fte coutable umber of dstct possble values We ca assume that the values ca be lsted or couted Radom Varables that fall alog pots o a terval, ad ca t be fully couted, are call Cotuous Radom Varables How ca you tell t s a dscrete radom varable? Coutable Usually s descrbed as the umber of Teds to be whole umbers Number of studets applyg to a uversty Number of errors o a test Number of bactera per cubc cetmeter of water Number of heart beats of a patet To descrbe a dscrete radom varable Specfy all the possble values t ca assume Assg correspodg probabltes to each value Specfyg a dscrete radom varable The umber of Shelds whe tossg two cos Let =umber of Shelds observed Sheld Sheld SS Sheld Number SN Number Sheld NN Number Number NN So takes o the followg values of Shelds O NN SN or NS SS Specfyg a dscrete radom varable Tossg two cos ad otg the umber of Shelds We ca a pror assg probabltes to Number of Heads Sample Pots NN SN NS SS p( Probabltes. *. =. (. *.) =.. *. =.

4 Specfyg a dscrete radom varable Tossg two cos ad otg the umber of Shelds Probablty Dstrbuto: P( Ths completely defes the dscrete radom varable Coectg probabltes to the values results the probablty dstrbuto The probabltes are dstrbuted over the values Ca be show by a Graph Table Formula wll come later Specfes the probablty assocated wth each value Requremets: P( ad for all values of x P( = Probablty Dstrbuto: P( for Flppg Two Cos ad Notg the Number of Shelds Notato P(x = ) = P(x = ) = P(x = ) = P(x < ) = P(x > ) = Probablty Dstrbutos of Dscrete Radom Varables Sometmes the probabltes are kow a pror Sometmes they are observed expermets, a posteror Ad sometmes we supply the probabltes based o subjectve formato or a smulato (a what f scearo) The maager of a large computer etwork has developed the followg probablty dstrbuto of the umber of terruptos per day The data ad assged probabltes based o past experece Iterruptos ( P( Ope up Excel ad eter these umbers a worksheet Graph t usg Iset, Graph, Bar Chart P Probablty Dstrbuto of Network Iteruptos 6 # of Iteruptos

5 Mea ad Varace of a Dscrete Varable We ca thk of our observed probablty dstrbuto for x as havg a mea ad varace A expected value s aother term for the mea whe dealg wth a probablty dstrbuto Expected Values of Dscrete Radom Varables The expected value of a dscrete radom varable s E( = x p( x ) = µ The sum of each value tmes the probablty of that value The Expectato of a Dscrete Varable. I take each value of the dscrete varable. Multply by the probablty assocated wth the value. Sum the calculatos E( = x p( x ) = µ Use your Excel fle to calculate the compoets of the expected value eter a ew colum of x* P( Excel example of etwork dsruptos Iterruptos ( P( x*p( Sum..7 Expected Value The Varace of a Dscrete Radom Varable The varace of a dscrete radom varable s gve as the Expectato of the squared devatos about the populato mea: E [( x µ ) ] = ( x µ ) p( x ) = σ Sum of the squared devato of each value from the mea tmes the probablty of the value The Varace of a Dscrete Varable. I take each value of the dscrete varable. Subtract the mea. Square the result. Multply by the probablty assocated wth the value. The sum each of these calculatos E[( x µ ) ] = ( x µ ) p( x ) = σ Use Excel to create aother colum of data for the varace

6 Compoets of the Varace Iterruptos ( P( x*p( Var Calc Mea ad Varace of Dscrete Radom Varable Remember, we sad the varace s the mea squared devato about the mea The stadard devato s the square root of the varace I our example, σ =. Sum Varace - Expectato P() P() If I radomly selected detsts, how may would I expect to use laughg gas? E()= (.) + (.768) + (.) + (.6) + (.9) + (.778) E() =. - Expectato P() P() If I radomly selected detsts, what s the varace (σ )? σ = (-) (.) + (-) (.768) + (-) (.) + (-) (.6) + (-) (.9) + (-) (.778) σ =.998 σ =.9 =. 6

7 .. Probablty Dstrbuto of the Dscrete Varable Probablty Dstrbuto of E() = σ =. σ =. Smplfed formula for the Varace σ = ( x P( ) µ p().. Number of Detsts. Square each x value. The multply by P(. Add them all together. The subtract µ Portfolo Expected Retur Portfolo Example Portfolo data uder dfferet market codtos ca be thought of as a Dscrete radom varable We desgated expected returs ( uder dfferet scearos, ad the assg probabltes to these outputs A vestor plas to vest Dell Computers or gold. She fgures probabltes uder four market codtos depresso, recesso, stable, growth She the assgs probabltes to each scearo, ad the estmated returs for Dell stock or gold. The returs are percet crease value. Eter ths data to a ew Excel worksheet Market Codtos Probablty Dell Retur Gold Retur Depresso. -.. Recesso... Stable.. -. Growth...9 Calculate the mea, varace, ad stadard devato for each vestmet strategy Leave the data ths form ad use other cell areas to make the calculatos Portfolo Aalyss Whe I am terested fdg a optmal vestmet strategy betwee two vestmet strateges I must also take to accout the Covarace of the two radom varables. The covarace takes cout how x ad y vary about ther meas together, weghted by ther jot probabltes N σ = xy [ x E( ][ y E( y) ] P( x y ) 7

8 Portfolo Aalyss Expected Value of the Sum of Two Radom Varables E ( x + y) = E( + E( y) Varace of the Sum of Two Radom Varables Var( x + y) = x+ y σ = σ + σ + σ x y xy Portfolo Aalyss Expected Retur s based o the expected values of each vestmet tmes the weght (w) gve to that vestmet the portfolo (.e., how much s vested each strategy ( w) E( + ( w) E( ) E( P) = y Portfolo rsk s the stadard devato of the Retur σ = w σ + ( w) σ + w( w) σ P x y xy PHStat wll do all of ths for you! Decso-Makg Covarace ad Portfolo Maagemet Number of outcomes Check the Portfolo Maagemet Aalyss You ca copy the labels ad your data to the worksheet. PHStat wll calculate these values for you check your expected values ad varaces 8