CA 121 Intro to Programming

1
Random Numbers
What is a random number? I can ask you to give
me a random number between 1 and 100. Suppose
you respond with 86. How did you choose that
particular number? What algorithm did you use?
Suppose I ask you for another random number. How
will you choose that one? How can we, as
programmers, instruct the computer how to
calculate a sequence/stream of random numbers?
2
Random Numbers
Sometimes we, as programmers, need to generate
random numbers. For example, in the Birthday
Problem, we need to generate a random birthday
for each person. We can do this simply by
generating a random number between 1 and 365,
inclusive. We can ask the computer to give us a
random number between 0.0 and 1.0, 0.0, 1.0),
and then we can scale the number to get it into a
range that we require. In Visual Basic, the
function Rnd() will return a number between 0.0,
1.0).
3
Random Numbers

• So, if we want a random integer between 1 and
  365, we can do the following
• Dim birthday As Integer
• birthday int( Rnd() 365 ) 1
• Lets look at the right-hand side above in
  detail
• Rnd() gives us a random number between 0.0,
  1.0)
• Multiplying by 365 gives us a number between
  0.0, 365.0)
• Conversion to int gives us an int between 0,
  364
• Finally, adding 1 gives us an int between 1,
  365.

4
MidSquare Method
There are many ways to generate random numbers.
Well take a closer look at the Midsquare method
of generating random numbers between 0.0,
1.0). First, we start with a four-digit seed
value, for example, 7182. We then square it to
get a number up to eight digits long. If the
number has fewer than eight digits, we pad the
left with zeroes until we get eight digits. In
our example, 71822 gives us 51581124. Now we
choose the middle four digits of our result,
which is 5811 in our example. We divide by
10,000 to get our first random number, e.g.
0.5811. We repeat this process indefinitely
5
MidSquare Method
i zi (zi)2 Middle four digits Pseudo-random num
0 7182 51581124 5811 0.5811
1 5811 33767721 7677 0.7677
2 7677 58936329 9363 0.9363
3 9363 87665769 6657 0.6657
4 6657 44315649 3156 0.3156
5 3156 09960336 9603 0.9603
6 9603 92217609 2176 0.2176
7 2176 04734976 7349 0.7349
6
Pseudo-random Numbers
It is important that the stream of pseudo-random
numbers that we generate appear random, i.e.
there is no discernable pattern to the numbers.
Consider the stream generated by our previous
example 0.5811, 0.7677, 0.9363, 0.6657,
0.3156, 0.9603, 0.2176, 0.7349, Can you find
any pattern in the numbers? Even though the
stream is actually computed using a formula, and
well get the same stream of numbers if we start
with the same seed value, the pseudo-random
numbers are good enough to use if they appear
random, as determined by various statistical
tests that are beyond the scope of this course
7
Midsquare code
'seed is a four-digit starting value (1000 lt
seed lt 9999) 'n is the number of random
variates to generate (n gt 0) Dim n, seed As
Integer inputs to read from user Dim i, sqr,
mid As Integer System.Console.Write("How many
random numbers do you require? ") n
System.Console.Readline() System.Console.Write(
Enter a 4-digit seed (between 1000 and 9999)
") seed System.Console.Readline() mid
seed for i 1 to n sqr mid
2 'square the four-digit number Now
choose the "middle 4" digits of sqr sqr
sqr \ 100 'lose the right-most 2 digits
mid sqr Mod 10000 'keep the new right-most
4-digits System.Console.Writeline(mid /
10000.0) next i