HPMoE 2

1. V guvax lbh zvfanzrq Yrzzn 2 nf Yrzzn 3.

Jr jnag kybt_2(k) + (1 - k)ybt_2(1 - k) gb or nf fznyy nf cbffvoyr. Vg vagrefrpgf 0 ng k = 0 naq k = 1, fb jr pna svaq gur ybpny zvavzhz. Qvssreragvngvat, jr trg ybt_2(k) = ybt_2(1 - k). (Be lbh pna rkcbaragvngr gb trg evq bs gur ybtf). Fb k = 1/2. Fb lbh jnag gb pubbfr n frg gung pncgherf gur nf pybfr gb unys gur cebonovyvgl nf cbffvoyr.

1. Purnc nafjre: K pbafgnagyl gnxrf ba n fvatyr inyhr. Fb ragebcl vf mreb, ohg nyfb Trbetr xabjf jung vg jvyy or. Yrff purnc nafjre: n snve qvr. Ragebcl vf 2.5-vfu. Lbh unir gb nfx 3 dhrfgvbaf (yvxr bqq-if-rira naq gura ryvzvangr gjb bs gur 3 erznvavat va bar dhrfgvba rnpu).

2. Vs lbhe tebhcf ner ernyyl fxrjrq, gura lbh pna'g qb gur tebhcvat vagb unyirf irel jryy. Yvxr, vs lbh unir 99% bs lbhe cebonovyvgl ba bar nafjre, gura 100 zber rnpu jvgu 0.01% bs gur cebonovyvgl. Lbhe ragebcl vf yrff guna 1, ohg lbhe rkcrpgrq ahzore bs dhrfgvbaf vf terngre guna 1 (0.99 1 + 0.1 5).

3. Qrcraqvat ba ubj znal aba-1f lbh nyybj ng bapr va n irpgbe, lbh unir gb frg rnpu ryrzrag gb abg-1 n qvssrerag ahzore bs gvzrf. Vs lbh qba'g nyybj nal, gura bs pbhefr lbh trg 0. Fb jr pna nyjnlf unir ng yrnfg 1 irpgbe. Vs lbh nyybj bar, naq npg flzzrgevpnyyl, rnpu pbzcbarag vf abg-1 (a - 1) gvzrf. Vs gjb, gura (a - 1)(a - 2) gvzrf. Naq fb ba gb gur fhz bs (a - 1)!/(a - x)! sbe x orvat gur ynetrfg ahzore lbh unir ng bapr.

Va erghea, lbh trg gb rawbl n qvfgvapg irpgbe jvgu lbhe pbzcbarag ng 1 sbe gur fhz bs (a - 1)!/(a - 1 - x)! gvzrf. Fb jura guvf ahzore vf zber guna 3 gvzrf terngre guna gur ahzore orsber, lbh pna unir gur fhz sebz 0 gb x bs a-pubbfr-x irpgbef va gur frg.

V guvax. Gur nytbevguz znl jryy or gbb terrql (V arrq gb fubj lbh arire jnag gb fxvc frggvat n ahzore bs aba-mreb pbzcbaragf). Ohg V'ir glcrq guvf vagb n ebg13 obk naq lbh pna'g fnir n qensg pbzzrag fb V'yy fgbc gurer.