CO 745, Mdterm Len Cabrera. A multle choce eam has questons, each of whch has ossble answers. A student nows the correct answer to n of these questons. For the remanng - n questons, he checs the answers comletel at random. To tae nto consderaton the chance of students randoml selectng the correct resonse, the rofessor has decded that for each correct answer the student gets one ont, and for each wrong answer he gets -/ onts. a Derve an eresson for the denst functon for the students score. b What s the eected value of the total score for the test? Do ou have an recommendatons that ou could mae? a Probablt of gettng uncertan queston correct s / umber of uncertan questons student gets correct s bnomal dstrbuton wth arameters - n # of uncertan questons and / robablt of success Let X number of uncertan questons the student gets correct ~ Bn - n, / n n f, X n, Var X n umber of uncertan questons student gets ncorrect s - n - X Score equals number of certan questons n lus number of uncertan questons student gets correct X, mnus / tmes number of uncertan questons student gets ncorrect - n - X Therefore, score s a functon of X: SX n X - / - n - X n X - - n/ X/ S n n, where - n That s, the student gets n onts and we automatcall subtract / onts for all uncertan questons, then add / onts for the uncertan questons he gets rght. Double chec mn and ma scores: S n - / - n gets none of the uncertan questons rght S - n n - / - n - n / - n gets all uncertan questons rght SX s a strctl ncreasng functon of X, so there s a unque that gves a artcular score s. n n The denst of the score s g s s n n, where. Ths s vald for n - / - n s b The score s a lnear functon of X, therefore, the eected value of the functon s equal to the functon of the eected value,.e., SX SX Substtute the value for X gven above nto the formula for SX: S X n n n n n deall, the eected value of the score would be n n order to measure eactl what the student new. Unfortunatel, even b subtractng / onts, the eected value wll be greater than n although less than t would be wthout subtractng the onts. Also, - n/ wll never be zero. n order to get t closer to zero, the rofessor should mae as large as ossble.e., more choces for each queston. of 8
f enalt of / - for ncorrect answers, then S n.. umbers are selected at random from the nterval,. a f numbers are selected, what s the robablt that eactl 5 of them are less than /? b f numbers are selected, on average how man of them are less than /? c f numbers are selected, what s the robablt that the average of the numbers s less than /? a Unform, / / Pr[Un, < /] d ] / X # Un, samles < / ~ Bnomal wth n and / 5 f 5 5 5 6 / 56.46 5 b X / 5 Double checed wth smulaton n cel: Trals: Average: 5. StDev:.56 F5.6 F4.58 Pr[5 samles < /] F5 - F4.5 c Use Theorem: f random samle s from a oulaton wth mean µ and varance, then the samle mean s a random varable wth mean µ and varance /n. n ths case, oulaton mean and varance are from Un,: X /, VarX / average of samle of Un,; ~ /, / / / / Pr [ < / ] Pr < / / Double checed wth smulaton n cel: Trals: Average:.5 StDev:. Pr [ z < ].5 Count Prob. Percent Prob. 5 5 5.8.6.4. PDF of umbers 4 5 6 7 8 9 CDF of umbers 5 7 9..8.6.4...8.6.4..8.6.4 PDF of Avg of umbers.4.4.45.48.5.5.55.58 CDF of Avg of umbers..8.4.48.5.58 of 8
. Let,,..., m be a random samle from a normal dstrbuton wth mean µ and varance, and,,..., n be a samle from a normal dstrbuton wth mean µ and the same varance. Also and j are ndeendent for all, j. Let and be the samle means, and s and s be the samle varances. a Wrte down the dstrbuton of z b What s the dstrbuton of, t's mean and varance. m s n s u c Usng z and u, construct a random varable that has a t-dstrbuton. What are ts degrees of freedom? a Theorem: f random samle s from a oulaton wth mean µ and varance, then the samle mean s a random varable wth mean µ and varance /n. ~ µ, /m and ~ µ, /n z ~ µ µ, 87 of Statstcs for ngneerng and the Scences. rd ed. Mendenhall & Sncch f z e z µ µ π z µ µ Var z b m s n s ~ χ m and ~ χ n 5 of Statstcs for ngneerng and the Scences. rd ed. Mendenhall & Sncch From class notes: f X ~ χ and X ~ χ m and the're ndeendent, then X X ~ χ m n u m s n s m ~ χ m n f u u m n m n e u m n Γ u m n Var u m n of 8
z c From class notes: z ~, and X ~ χ n t ~ t n X / n z ~ z µ µ µ µ, z ' ~, z' ~ t m n u / m n 4. Suose that the random varable X has a Pareto denst f ;, / / for, > and a Derve the denst of the random varable Y lnx. b After etensve testng to be done n the second half of the course ou have determned that. Gven ths, fnd the Y and VarY. c Gven our answer to b, what s Y and VarY for the general case where s not equal to? a Y lnx X ds e d g Y e s ds f s / e d e e, ln b g e, ~ onental Y Var Y c Snce s a constant, Y and Var Y 4 of 8
5. Suose that X s random varables wth multnomal denst: f,, ;,,,, a Gven a random samle of observatons, wrte down the lelhood and log-lelhood functons. b What are the mamum lelhood estmators of and? c Show that the mamum lelhood estmators, ˆ and ˆ, are unbased estmators of and. d Derve the nformaton matr for and. e What are the Cramer-Rao lower bounds for Var and Var? a L ; f,, where,, and, L ; ln L ; ln ln ln For smlct let,, b To fnd ML, tae artals of ln L ; wrt, set them equal to zero and solve for ln L ; ln L ; Solve qn for b frst gettng common denomnators Get rd of the denomnator b multlng both sdes b the denomnator Combne terms for Solve for b tang t to the other sde and dvdng b ow wor on qn to solve for ; frst get common denomnators As before, get rd of the denomnator b multlng both sdes b the denomnator 5 of 8
Combne terms for ow substtute qn for Brea u fracton so 's cancel Combne terms wth on one sde and everthng else on the other. Get common denomnators on the left term for and multl out the term on the rght. Multl both sdes b and notce the terms cancel ow solve for. We can cancel the and substtute ˆ ow substtute qn 4 nto qn to fnsh solvng for / ˆ Substtute ˆ We sweat off brow... 4 ˆ ˆ c ˆ ˆ unbased unbased 6 of 8
7 of 8 d, where j j L ; ln, To mae thngs qucer and easer to read: e CRLB s [ ] Usng d c b a a c b d cb ad
8 of 8 ad cb [ ] Var Var