Question

3. Generate random numbers as indicated and comment on the results. (a) generate 10 random numbers using the middle-square method using zo = 1009. (b) generate 10 random numbers using linear congruence with a = 5,b=1 and c= 8.

Answer 1

Pq No-1 Solution : - Given data : A). Generate 10 Random numbers using the middle - Square Method using 80 = 1009 : In the middle Square method we have to start with the u-digit numbers, is called Seed → In Every Step we have to Square our Seed and than Next Number of the Sequence will be middle Four digits af Square number. xo= 1009. → X0 = (x0)2 = (1009) - 01918081 2 = 0180 → (,)? = (01802 - 00032400 82.=.0324

Pq No- >02) - (0324) 2 - 00104976 x₃ = loug → (x3)} = (1049) - 01100401 24 = 1004 bu)? = (100u)? - 01008016 25 - 0080 → (25)2 = 60080) = 00006400 *= 10064) = (0o6w? > (346)2 - 00004096 27-0040 → (82)2 = (004032 - 00001600 28 = 0016

Pq No - 6 → (X3)? - (0016) 200000 256 Hq = 0002 → (38q)2 = (2002) - 00000004 810= 0000 Now all the an, n=10 / xn=0000/ - with Seed is 1009 our Sequence is Converges too fast after 25 & rt get it's limit at dio is 0000 B). Generate 10 Random number using linear Congruence with Q=5 b=1 c=8.. there we have to start with a number to is Called Seed than wext term is calculated by the formula Xn+1=(ax+b ) (mode)

► 2,= (5(0) +1) mod 8 = 6 (mod 8) = 6 → 82.-(5(6)+ 1) mods = 31 (mods) 7 → 43 = (5(1) +1) mods = 36 (mod 8) → xy = (5(4)+ 1) mod 8 - ! (mod 8) - 5 → 85=(5 (5)+1) mods Q6 (mod 8) ->86 = (5 (2)+1) mod 8 = 11 (mod 8) = 3 → 87 = (5 (3)+1) mod 8 = 16 (mod 8) - 0

la no-6 > dle = (500) + 1 ) mod 8 - 1 (mod 8) → 89 = (50)+1 ) mod 8 = 6 (mod 8) we Start Repeated there so x 10 =*2=7 *1 = 23=4 So-- on is Tence, Started there wo took (mod 8) So our Sequence at the Representation of 8 Steps.