For each of the following determine the constant c so that f

For each of the following, determine the constant c so that f(x) satisfies the conditins of being a probability mass function (p.m.f.) for a random variable X.

a. c/(x+2)(x+3);x = 0,1,2,3

b. c(1/5)^x;x = 1,2,3,...

c. cx^2;x = 1,2,3,...,n

Solution

a)

Note that

c/[(x + 2)(x + 3)] = c[1/(x + 2) - 1/(x + 3)]

Thus,

c[1/(0 + 2) - 1/(0 + 3)] + c[1/(1 + 2) - 1/(1 + 3)] + c[1/(2 + 2) - 1/(2 + 3)] + c[1/(3 + 2) - 1/(3 + 3)] = 1

Cancelling terms,

c[1/2 - 1/6] = 1

c[1/3] = 1

c = 3 [ANSWER]

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b)

This is a geometric series with first term a1 = c(1/5) with common ratio r = 1/5.

Thus, the sum is

Sum = a1 / (1 - r) = c(1/5) / (1 - 1/5) = c/4

Thus,

c/4 = 1

c = 4 [answer]

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The sum of the first n squares is = n(n+1)(2n+1)/6

THus, including the factor c,

cn(n+1)(2n+1)/6 = 1

Thus,

c = 6/[n(n+1)(2n+1)] [answer]