MAT 263 PROBABILITY THEORY [Teori Kebarangkalian]

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1 UNIVERSITI SAINS MALAYSIA Second Semeser Eaminaion 04/05 Academic Session June 05 MAT 63 PROBABILITY THEORY [Teori Kebarangkalian] Duraion : 3 hours [Masa : 3 jam] Please check ha his eaminaion paper consiss of EIGHT pages of prined maerial before you begin he eaminaion. [Sila pasikan bahawa keras peperiksaan ini mengandungi LAPAN muka sura yang berceak sebelum anda memulakan peperiksaan ini.] Insrucions: Answer TEN (0) quesions. [Arahan: Jawab SEPULUH (0) soalan.] In he even of any discrepancies, he English version shall be used. [Sekiranya erdapa sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai.] /-

2 [MAT 63]. An elecronic componen is composed of 3 ransisors, each of which is eiher nondefecive or defecive. A echnician wans o observe he saus of each ransisor. Le he oucome be given by he vecor (,, 3), where i is equal o if ransisor i is non-defecive and is equal o 0 if ransisor i is defecive. (i) (iii) Specify he sample space, S of he eperimen. The componen will funcion if all ransisors are non-defecive, or if ransisors and 3 are boh non-defecive, or if ransisors and 3 are boh non-defecive. Le A be he even ha he componen will funcion. Specify all he oucomes in A. Le B be he even ha ransisors or is defecive. Specify all oucomes in B. (iv) Specify all he oucomes in he even A B. [0 marks]. Suau komponen elekronik erdiri daripada 3 ransisor, yang masing-masing sama ada idak rosak aau rosak. Seorang jurueknik ingin memanau saus seiap,,, yang mana ransisor. Biarkan kesudahannya diberikan oleh vekor ( ) 3 i sama dengan jika ransisor i adalah idak rosak dan sama dengan 0 jika ransisor i rosak. (i) Tenukan ruang sampel, S bagi eksperimen ersebu. (iii) Komponen ersebu akan berfungsi jika semua ransisor idak rosak, aau jika ransisor dan 3 kedua-duanya idak rosak, aau jika ransisor dan 3 kedua-duanya idak rosak. Biarkan A sebagai perisiwa komponen akan berfungsi. Tenukan semua kesudahan dalam A. Biar B sebagai perisiwa yang ransisor aau rosak. Tenukan semua kesudahan dalam B. (iv) Tenukan semua kesudahan dalam perisiwa A B. [0 markah] 3/-

3 3 [MAT 63]. Le X be a coninuous random variable wih probabiliy densiy funcion (p.d.f.) given by c( ), 0 < c ( ), < f( ) = c, < 3 / 6 0, elsewhere. (i) Deermine he consan c. (iii) Deermine he cumulaive disribuion funcion (c.d.f.), F(). If X, X and X 3 are hree independen observaions from X, wha is he probabiliy ha eacly one of hese hree numbers is greaer han? [30 marks]. Biar X sebagai pembolehubah rawak selanjar dengan fungsi keumpaan kebarangkalian (f.k.k) diberi oleh (i) Tenukan pemalar c. c( ), 0 < c ( ), < f( ) = c, < 3 / 6 0, di empa lain. (iii) Tenukan fungsi aburan longgokan, F(). Jika X, X dan X 3 ialah iga cerapan ak bersandar daripada X, apakah kebarangkalian bahawa epa sau daripada keiga-iga nombor ini lebih besar daripada? [30 markah] 3. A random variable X has a Poisson disribuion wih parameer λ. A random variable, Y also has a Poisson disribuion and he probabiliy ha Y = i is given by PY ( = i) = P( X= i X> 0). Find E(Y). [0 marks] 3. Suau pembolehubah rawak X mempunyai aburan Poisson dengan parameer λ. Suau pembolehubah rawak, Y juga mempunyai aburan Poisson dan kebarangkalain bahawa Y = i diberi oleh PY ( = i) = P( X= i X> 0). Dapakan E(Y). [0 markah] 4/-

4 4 [MAT 63] 4. Le X be a random variable wih mean, µ and variance, σ. The momen generaing funcion (m.g.f.) of X is denoed by MX ( ), < <. Suppose c is a posiive consan and Y is a random variable wih m.g.f. cm ( X () ) MY () = e, < <. Find he mean and variance of Y in erms of µ and σ. [0 marks] 4. Biarkan X sebagai pemboleh ubah rawak dengan min, µ dan varians, σ. Fungsi penjana momen (f.p.m.) bagi X dilambangkan sebagai MX ( ), < <. Andaikan c suau pemalar posiif dan Y ialah suau pemboleh ubah rawak dengan cm ( () ) f.p.m. MY () = e, < <. Cari min dan varians bagi Y dalam sebuan µ dan σ. 5. A random variable R wih p.d.f. given by [0 markah] r (), 0 e r s f r = r > s 0, elsewhere is called a Rayleigh random variable and is said o have he Rayleigh disribuion wih parameer σ. Deermine and idenify a p.d.f. of he random variable Y = R. [30 marks] 5. Pembolehubah rawak R dengan f.k.k. diberi oleh r r (), 0 e σ f r = r > σ 0, di empa lain dipanggil pembolehubah rawak Rayleigh dan dikaakan mempunyai aburan Rayleigh dengan parameer σ. Dapakan dan kenalpasi suau f.k.k. bagi pembolehubah rawak Y = R. [30 markah] 5/-

5 5 [MAT 63] 6. The number of power banks sold weekly a a cerain shop lo is a random variable wih epeced value 0. (i) Give an upper bound of he probabiliy ha ne week s sales eceed 00. Suppose ha he variance of he number of power banks sold weekly is 40. Give a lower bound of he probabiliy ha he ne week s sales are beween 00 and 40. [0 marks] 6. Bilangan bank kuasa yang dijual di sebuah lo kedai seiap minggu adalah pembolehubah rawak dengan nilai jangkaan 0. (i) Beri baas aas kepada kebarangkalian jualan pada minggu hadapan melebihi 00. Andaikan varians bilangan bank kuasa yang dijual seiap minggu adalah 40. Beri baas bawah kepada kebarangkalian jualan pada minggu hadapan adalah anara 00 dan 40. [0 markah] 7. Suppose ha he join p.m.f. of random variables X and Y is given by y f( y, ) =, =, 0, ; y=,, (i) Verify ha f( yis, ) a join p.m.f. Compue PXY ( < ), PY ( X< ) and PX ( = 0 Y= ). [0 marks] 7. Andaikan f.j.k. ercanum bagi pembolehubah-pembolehubah rawak X dan Y diberi oleh y f( y, ) =, =, 0, ; y=,, (i) Sahkan bahawa f( yadalah, ) suau f.j.k. ercanum. Hiung P( XY < ), P( Y X < ) dan P( X = 0 Y = ). [0 markah] 6/-

6 6 [MAT 63] 8. Suppose ha X and Y are random variables. The marginal p.d.f. of X is f( ) = ke, > 0 { 0, elsewhere. Also, he condiional p.d.f. of Y given ha X = is y hy ( ) = e, y > 0 { 0, elsewhere. Deermine (i) he value of k. he marginal p.d.f. of Y. Are X and Y independen? Jusify your answer. (iii) EYX ( ) =. [30 marks] 8. Andaikan X dan Y adalah dua pembolehubah rawak. F.k.k. su bagi X adalah f( ) = ke, > 0 { 0, di empa lain. Juga, f.k.k. bersyara bagi Y diberi X = adalah y hy ( ) = e, y > 0 { 0, di empa lain. Tenukan (i) nilai k. f.k.k. su bagi Y. Adakah X dan Y idak bersandar? Tenusahkan jawapan anda. (iii) EYX ( ) =. [30 markah] 7/-

7 7 [MAT 63] 9. X, X,..., X 7 is a random sample of size n = 7 from he normal disribuion wih mean 5 and variance 0. The saisics X and S are defined as respecively. = 7 i= X X 7 i and i= S = ( Xi X) 7 Find P( X 4, 5 < S < 5) and P( 5 S 5 X 4) 6 < <., [0 marks] 9. X, X,..., X 7 ialah suau sampel rawak bersaiz n = 7 daripada aburan normal dengan min 5 and varians 0. Saisik X dan saisik S diakrifkan masingmasing sebagai = 7 i= X X 7 i dan i= S = ( Xi X) 7 Dapakan P( X 4, 5 < S < 5) dan P( 5 S 5 X 4) 6 < <.. [0 markah] 0. Le X, Y be wo independen sandard normal random variables. Show ha a random variable U = X has a Cauchy disribuion wih p.d.f. defined as Y f( u) =, < u<. π + u ( ) [Hin: Use an auiliary variable, V and deermine he marginal p.d.f. of U from he join p.d.f. of U and V, f( uv, ).] [40 marks] 0. Biar X, Y sebagai dua pembolehubah rawak normal piawai yang ak bersandar. Tunjukkan bahawa suau pembolehubah rawak U = X mempunyai aburan Y Cauchy dengan f.k.k. diakrif sebagai f( u) =, < u<. π + u ( ) [Peunjuk: Gunakan pembolehubah banu, V dan enukan f.k.k. su bagi U daripada f.k.k. ercanum bagi U dan V, f( uv, ).] [40 markah] 8/-

8 8 [MAT 63] Appendi Disribuion Probabiliy Densiy Funcion Momen generaing funcion Bernoulli p ( p), = 0,, 0 < p < pe + q n Binomial p ( p n ), = 0,,..., n, 0 < p < Hypergeomeric ( pe + q) n n n r, = 0,,..., r n or =,,..., n r n+ n r pe Geomeric ( p) p, =,,... qe Negaive Binomial r r p ( p), = r, r+,... pe r qe r Poisson e! λ λ, = 0,,,..., λ > 0 e λ ( e ) Uniform, α < < β β α ( µ ) Normal ep, < < pσ σ Eponenial λ λe, 0 β α e e, 0 (, if = 0) ( β α) e µ + σ λ, < λ λ Gamma Chi-square Bea λ λ λ α Γ( α) Γ ( α + β) Γ α λ ( ) e, 0, > 0, > 0 r e, 0 r / Γ( r / ) ( ), 0 < <, α > 0, β > 0 ( ) ( ) α β α Γ β α λ λ r /, < λ, < - ooo O ooo -

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