213 For each of the following determine the constant c so th

2.1-3. For each of the following, determine the constant
c so that f (x) satisfies the conditions of being a pmf for
a random variable X, and then depict each pmf as a line
graph:

(e) f (x) = x/c, x = 1, 2, 3, . . . , n.
(f) f (x) = c/[(x + 1)(x + 2)], x = 0, 1, 2, 3, . . . .
Hint: In part ( f ), write f (x) = 1/(x + 1) 1/(x + 2).

Solution

e)

1/c + 2/c + 3/c ... + n/c = 1

which is an arithmetic series of n terms, with common difference 1/n.

Thus,

(1/c)(1 + n)(n)/2 = 1

Thus,

c = n(n+1)/2 [answer]

************

f)

Note that we can write

f(x) = c[1/(x + 1) - 1/(x + 2)]

Thus,

= c{[1/(1 + 1) - 1/(1+2)] + [1/(2 + 1) - 1/(2+2)] + [1/(3 + 1) - 1/(3+2)] + ...}

= c{[1/2 - 1/3] + [1/3 - 1/4] + [1/4 - 1/5] ...}

As you can see,only the 1/2 will survive, as the succeeding fractions cancel with the next term. Thus,

= c(1/2) = 1

Thus,

c = 2 [answer]