Extracted text: H 10:07 ••. o Define the cumulative sums Si = a1 + a2+...+ ai and So = 0. Pick a random integer r uniformly between 0 and Sn – 1. - o Find the unique index i between 1 and n such that S;-1
< s;.="" geometrically,="" this="" subdivides="" the="" interval="" [0,="" sn)="" into="" n="" subintervals="" [s;-1,="" s;),="" with="" the="" length="" of="" subinterval="" i="" proportional="" to="" a¡.="" for="" example,="" if="" the="" discrete="" distribution="" is="" defined="" by="" (а1,="" а2,="" аз,="" а4,="" аs="" ,="" а6)="" —="" (10,="" 10,="" 10,="" 10,="" 1="" then="" the="" cumulative="" sums="" are="" (s1,="" s2,="" s3,="" s4,="" s5,="" s6)="(10," 20,="" 30,="" 40,="" and="" the="" following="" diagram="" illustrates="" the="" 6="" subintervals:="" pick="" a="" random="" integer="" r="" between="" 0="" and="" s,="" -="" 1.="" 1="" if="">< 10="" 4="" if="" 30="">< 40="" 6="" if="" 50="">< 100="" 10="" 20="" 30="" 40="" 50="" 100="" so="" s2="" s3="" ss="" s6="" in="" probability="" theory,="" this="" is="" known="" as="" sampling="" from="" a="" discrete="">
Extracted text: (€) 10:06 ••. 1. Discrete distribution. Write a program DiscreteDistribution.java that takes an integer command-line argument m, followed by a sequence of positive integer command-line arguments a1, a2,.. random indices (separated by whitespace), choosing each index i with probability proportional to a¡. An, and prints m 9 •• • -/Desktop/arrays> java DiscreteDistribution 25 1 1 1 111 5 2 4 4 5 5 4 3 4 3 1 5 2 4 2 6 1 3 6 2 3 2 4 1 4 ~ /Desktop/arrays> java DiscreteDistribution 25 10 10 10 10 10 50 3 6 6 16 6 2 4 6 6 3 6 6 6 6 4 5 6 2 2 66 2 6 2 -/Desktop/arrays> java DiscreteDistribution 25 80 20 1 2 1 2 1 1 2 1 1 1 1 1 1 11 2 2 2 1 111111 ~/Desktop/arrays> java DiscreteDistribution 100 301 176 125 97 79 67 58 51 46 6 2 4 3 2 3 3 1 7 1 1 3 47 1 4 2 2 1 1 3 1 8 6 2 1 3 6 185 1 3 6 1 1 2 3 8 7 4 6 4 3 1 5 3 3 7 3 1 3 177 2 2 3 6 5 4 1 1 1 7 2 3 5 2 2 1 4 1 2 1 2 1 2 2 3 2 8 4 3 2 1 8 3 5 3 3 8 1 2 3 3 1 2 3 1 To generate a random index i with probability proportional to a;: o Define the cumulative sums S; = a1 + a2 +...+ a; and 0. So Pick a random integer r uniformly between 0 and Sn – 1. o Find the unique index i between 1 and n such that S;-1 < r="">< s¿. - geometrically, this subdivides the interval [0, sn) into n subintervals [s;-1, s;), with the length of subinterval i proportional to aj. for example, if the discrete distribution is defined hy. s¿.="" -="" geometrically,="" this="" subdivides="" the="" interval="" [0,="" sn)="" into="" n="" subintervals="" [s;-1,="" s;),="" with="" the="" length="" of="" subinterval="" i="" proportional="" to="" aj.="" for="" example,="" if="" the="" discrete="" distribution="" is="" defined="">