Week 06 Discussion Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 8
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1 STAT W 6 Discussion Fll If h momn-gnring funcion of X is M X ( ), Find h mn, vrinc, nd pmf of X.. Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) 8 7, f ( ),,, 6, 8,. ( possibl vlus of X r vn non-ngiv ingrs:,,, 6, 8, ). Rcll W Discussion Problm (): his is vlid probbiliy disribuion. ) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? b) Us M X ( ) o find E ( X ).. Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) ln, f ( ) ln!,,,,. ( possibl vlus of X r posiiv ingrs:,,,, ). Rcll W Discussion Problm (b): his is vlid probbiliy disribuion. ) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? b) Us M X ( ) o find E ( X ).
2 . Suppos h momn-gnring funcion of X is M X ( ) ) Find E ( X ). b) Find SD ( X ).. Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) P ( X ),,,,, 6,, zro ohrwis. ) Find h vlu of h ms his is vlid probbiliy disribuion. b) Find P ( X is vn ). c) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? d) Find E ( X ). 6. L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) C, >, zro ohrwis. ) Find h vlu of C h would m f ( ) vlid probbiliy dnsiy funcion. b) Find h cumuliv disribuion funcion of X, F ( ) P ( X ). Hin : Should b F ( ), F ( ). c) Find h probbiliy P ( 6 < X < ). f) Find h 8h prcnil of h disribuion of X,.8. g) Find h pcd vlu of X, E ( X ). h) Find h sndrd dviion of X, SD ( X ).
3 7. L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) C,, zro ohrwis. ) Find h vlu of C h would m f ( ) vlid probbiliy dnsiy funcion. b) Find h probbiliy P ( X < ). c) Find h probbiliy P ( X > 7 ). d) Find h mn of h probbiliy disribuion of X. ) Find h mdin of h probbiliy disribuion of X. 8. Suppos rndom vribl X hs h following probbiliy dnsiy funcion: f ( ) cos, < < π, zro ohrwis. ) Find P ( X < π ). b) Find E ( X ). c) Find h mdin of h probbiliy disribuion of X.. L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) 6 ( ), < <, zro lswhr. Compu P ( < X < + ).. Suppos rndom vribl X hs h following probbiliy dnsiy funcion: f ( ), < <, zro ohrwis. ) Find P ( X < ). b) Find E ( X ). c) Find h momn-gnring funcion of X, M X ( ).
4 . L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f c,, 8, ohrwis. ) Find h vlu of c h ms f ( ) vlid probbiliy dnsiy funcion. b) Find h probbiliy P ( X < ). c) Find h mdin of h probbiliy disribuion of X. d) Find h mn of h probbiliy disribuion of X. ) Find h vrinc of h probbiliy disribuion of X.
5 Answrs:..-.- If h momn-gnring funcion of X is M X ( ), Find h mn, vrinc, nd pmf of X. f ( ), f ( ), f ( ). M X' ( ) 6. E ( X ) M X' ( ). OR E ( X ) ( ) + ( ) + ( ). M X'' ( ) 8. E ( X ) M X'' ( ) OR.8. E ( X ) ( ) + ( ) + ( ).8. Vr ( X ) E ( X ) [ E ( X ) ].8.8.
6 . Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) 8 7, f ( ),,, 6, 8,. ( possibl vlus of X r vn non-ngiv ingrs:,,, 6, 8, ). Rcll Discussion # Problm (): his is vlid probbiliy disribuion. ) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? M X ( ) E ( X ) Mus hv < for gomric sris o convrg. < ln. b) Us M X ( ) o find E ( X ). Rcll W Discussion Problm (): E ( X ). M ' X ( ) 8, < ln. 8 E ( X ) M ' X ( ). 6
7 . Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) ln, f ( ) ln,,,,.! ( possibl vlus of X r posiiv ingrs:,,,, ). Rcll Discussion # Problm (b): his is vlid probbiliy disribuion. Hin : Rcll h!. ) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? M X ( ) p ( ) ( ln ) + ll ( ln ) + ln!, R. ln! ln + ln ln b) Us M X ( ) o find E ( X ). Rcll W Discussion Problm (b): E ( X ) ln.8. MX ' ln, E ( X ) X ln. M '
8 . Suppos h momn-gnring funcion of X is M X ( ) ) Find E ( X ). M ' X ( ) E ( X ) M ' X ( ) b) Find SD ( X ). M X " ( ) E ( X ) M X " ( ) Vr ( X ) E ( X ) [ E ( X ) ] SD ( X )..8. OR M X ( ) f ( ) f ( ) f ( ) f ( ) ( ) f ( ) E ( X ) f ( ).6. Vr ( X ) ( ) f ( ).. SD ( X )..8. OR Vr ( X ) f ( ).6.6..
9 . Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) P ( X ),,,,, 6,, zro ohrwis. ) Find h vlu of h ms his is vlid probbiliy disribuion. Mus hv ll f. firs rm bs. +.. <..68. No:, whr is h goldn rio. b) Find P ( X is vn ). P ( X is vn ) f ( ) + f ( ) + f ( 6 ) + f ( 8 ) firs rm bs.68.
10 c) Find h momn-gnring funcion of X, M X ( ). For which vlus of dos i is? M X ( ) E ( X ) bs rm firs, < < ln ln.8. d) Find E ( X ). M ' X ( ), < ln. E ( X ) M ' X ( ) OR E ( X ) E ( X )... 6 E ( X ) Thrfor, E ( X ) +.68.
11 6. L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) C, >, zro ohrwis. ) Find h vlu of C h would m f ( ) vlid probbiliy dnsiy funcion. Mus hv C d f d. C C. C 7. 7 b) Find h cumuliv disribuion funcion of X, F ( ) P ( X ). Hin : Should b F ( ), F ( ). F ( ) P ( X ) 7 u du 7 u,. c) Find h probbiliy P ( 6 < X < ). P ( 6 < X < ) 6 7 d OR P ( 6 < X < ) F ( - ) F ( 6 )
12 f) Find h 8h prcnil of h disribuion of X, P ( X.8 ) F (.8 ) π.8 OR P ( X.8 ).. π.8 7 d 7 π.8 π For fun: Mdin 6.. g) Find h pcd vlu of X, E ( X ). E ( X ) f d 7 d 7 d h) Find h sndrd dviion of X, SD ( X ). E ( X ) f d 7 d 7 d 7 7. Vr ( X ) E ( X ) [ E ( X ) ] SD ( X )
13 7. L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) C,, zro ohrwis. ) Find h vlu of C h would m f ( ) vlid probbiliy dnsiy funcion. C 7 7 C d C C. C. b) Find h probbiliy P ( X < ). P ( X < ) 7 8 d c) Find h probbiliy P ( X > 7 ). P ( X > 7 ) 7 86 d
14 d) Find h mn of h probbiliy disribuion of X. E( X ) f d d ) Find h mdin of h probbiliy disribuion of X. F ( ) P ( X ) f ydy y dy y 7 7, 7 <. F ( ) P ( X ), <. F ( ) P ( X ),. F ( m ). 7 m 7. 7 m m
15 8. Suppos rndom vribl X hs h following probbiliy dnsiy funcion: f ( ) cos, < < π, zro ohrwis. ) Find P ( X < π ). P ( X < π ) π π cos d sin sin π.77. b) Find E ( X ). π π E ( X ) cos d π sin cos.78. c) Find h mdin of h probbiliy disribuion of X. F ( ) P ( X ) π cos y dy sin, < <. Mdin: F ( m ) P ( X m ). sin m. m π.6. 6
16 . L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f ( ) 6 ( ), < <, zro lswhr. Compu P ( < X < + ). E ( X ) 6 d 6 d 6 d E ( X ) 6 d 6 d 6 d Vr ( X ) E ( X ) [ E ( X ) ] , P ( < X < + ).8 d
17 . Suppos rndom vribl X hs h following probbiliy dnsiy funcion: ) Find P ( X < ). f ( ), < <, zro ohrwis. P ( X < ) d.76. b) Find E ( X ). E ( X ) d. c) Find h momn-gnring funcion of X, M X ( ). M X ( ) d d,. M X ( ) d.
18 . L X b coninuous rndom vribl wih h probbiliy dnsiy funcion f c,, 8, ohrwis. ) Find h vlu of c h ms f ( ) vlid probbiliy dnsiy funcion. Mus hv d f. 8 c d c d c d c c 7 c. c c. b) Find h probbiliy P ( X < ). P ( X < ) 7 d d d 7 7 OR P ( X < ) P ( X ) d d.
19 c) Find h mdin of h probbiliy disribuion of X. F X ( ), <, 7 F X ( ) y dy, <, 7 F X ( ) y dy y dy F X ( ), 8. 7, < 8, F X ( ). < mdin < 8.. F X ( mdin ) mdin ( mdin ) 8. mdin
20 d) Find h mn of h probbiliy disribuion of X. E ( X ) f d d d 7.7. ) Find h vrinc of h probbiliy disribuion of X. E ( X ) f d d d Vr ( X ) E ( X ) [ E ( X ) ]
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