Answers to the problems on the practice exam:

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Math.random() produces random values uniformly distributed in the interval [0, 1); that's a given. 

The expression Math.random() * 7 produces values uniformly distributed in the interval [0, 7) 

So  Math.random() * 7  + 6  will produce values uniformly distributed in the interval [6, 13)

So the answer is: 5

A proof of sorts can be obtained as follows: 

-bash-4.1$ ls -ld Two.java
-rw-r--r-- 1 dgerman www 556 Jan 29 13:19 Two.java
-bash-4.1$ cat Two.java
import java.util.*;

public class Two {
  public static void main(String[] args) {
    Map<Integer, Integer> histogram = new HashMap<Integer, Integer>();
    for (int i = 0; i < Integer.parseInt( args[0] ); i++) {
      Integer a = Two.genValue();
      if (histogram.containsKey( a )) {
        histogram.put( a , histogram.get( a ) + 1 );
      } else {
        histogram.put( a , 1 );
      }
    }
    System.out.println( histogram );
  }

  public static int genValue() {
    int x = (int)(Math.random() * (12 - 5) + 6);
    return x;
  }
}
-bash-4.1$ javac Two.java
-bash-4.1$ java Two 10
{7=3, 8=1, 9=3, 10=1, 12=2}
-bash-4.1$ java Two 100
{6=12, 7=15, 8=17, 9=9, 10=16, 11=12, 12=19}
-bash-4.1$ java Two 1000
{6=138, 7=150, 8=132, 9=129, 10=145, 11=152, 12=154}
-bash-4.1$ java Two 10000
{6=1425, 7=1399, 8=1417, 9=1473, 10=1394, 11=1429, 12=1463}
-bash-4.1$ java Two 100000
{6=14307, 7=14233, 8=14310, 9=14364, 10=14185, 11=14329, 12=14272}
-bash-4.1$

 2. ----------------------------------------------------------( 2)------------------------

    !(x % 4 != 0 || !(x % 100 == 0 && x % 400 == 0))

is equivalent to: 

    ((x % 400) == 0)

But it's not equivalent to: 

    ((x % 100) == 0)

    ((x % 100) != 0)

    ((x % 400) != 0)

You can prove this with DeMorgan: 


    !(x % 4 != 0 || !(x % 100 == 0 && x % 400 == 0))

is the same as 

      x % 4 == 0 && (x % 100 == 0 && x % 400 == 0)

which is

      x % 4 == 0 && (x % 400 == 0)

which is

      x % 4 == 0 && x % 400 == 0

which is

      x % 400 == 0

To prove it with a program you could randomly generate integers x. 

Then feed them in all five expressions. 

You should obtain counterexamples in all but one case, hence the clue. 

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-bash-4.1$ cat Seven.java
class Seven {

  public static double f(double x) { return g(x) + Math.sqrt(h(x)); }
  public static double g(double x) { return 4 * h(x); }
  public static double h(double x) { return x * x + k(x) - 1; }
  public static double k(double x) { return 2 * (x + 1); }

  public static void main(String[] args) {
    double x1 = f(2);
    System.out.println( x1 );
    double x2 = g(h(2));
    System.out.println( x2 );
    double x3 = k(g(2) + h(2));
    System.out.println( x3 );
    double x4 = f(0) + f(1) + f(2);
    System.out.println( x4 );
  }
}
-bash-4.1$ javac Seven.java
-bash-4.1$ java Seven
39.0
400.0
92.0
62.0
-bash-4.1$

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