Find the two positive numbers that satisfy the given requirements:

"the sum is s and the product is a maximum."

We are given with two conditions and we label each positve number as x and y. The sum of the 2 numbers is x + y equal to S. The product is maximum.
The product is xy. To find the maximum

P = xy = x (S- x) = Sx - x2
dP = S - 2x = 0
hence x = S/2

y then is equal to S/2 also. 

P(X = 0) = 1

16 , P(X = 1) = 4

16 , P(X = 2) = 6

16 , P(X = 3) = 4

16 , P(X = 4) = 1

16 (1)

Notice that the denominators of the five fractions are the same and the numerators of the five

fractions are 1, 4, 6, 4, 1. The numbers in the numerators is a set of binomial coefficients.

1

16 =

4

0

1

16, 4

16 =

4

1

1

16, 6

16 =

4

2

1

16, 4

16 =

4

3

1

16, 1

16 =

4

4

1

16

We can then write the probability mass function as

Date: November 1, 2005.

1