Zdjęcie0108 4

.19 Solve tłie foliowi na linear programmiiig problem.

maximize ) Pj^tj

n

Jmi

subject to ^2 ⁽h^xj S fi

«*■>< 1    i ~ 1,2,.... n

Jj > 0    J = 1,2,.... n.

Here. tbe numbers pj, j « 1,2.....n, are positive and sum to one Tb*

same is true of the qj 's:

Żw-i

9j > 0-

Furtbermore (with only minor loss of genenlity). you may mhwm chat

CL<a<..

ai aa

Finally. the paraineter i ii i smali positive number. See Exvcim 1 3 fbc th* moti\ ation for this problem

13 Suppose chat Y it i random \inible taking on one of u fcnown vałue*

et.02.... ,a_n.

Suppose we knou that V' eicher ha* dismbuuon p gi\ en by

P{Y afljlsp, or it has disrnbonon </ given by

F(K a Oj) * ąj.

Of course. che numbers Pj. j = l,2. ...,n are uonnegame and u«u to one. The same is true for the ąj *s. Based on a single obsanation ot > . we wiśh to guess whethei it has distnbuCLou p or dismbuuon q That u Ux each possible outcome aj. we will assert with probabUity s_t char the d&ttnburiao is p and with probability l-x_; that the dismbution u«/ We w uh

to detennme the probabilities x_}. j = 1.2.....a, such that che ptobebthry

of sayrng the dtstnbunon u p when m tacr u is </ has p*ob*b&hr> no largo te * uheie J u sorne smali posuive vaiue (such a* U U6) turthenuor* 0tven thu wuuwraint we w uh to inaximize the probabtlity that ara «a> the dMiiUnwa u p whaa m tad n is p FonmUate thu maxmiłzatw«*i ptobiean h a hneai pio^ótuuuu^ problem.