Question

Determine whether the given function can serve as the probability distribution (probability mass function) of random variable with given range: (a) f(x) = x−2/5 for x = 1, 2, 3, 4, 5 (b) f(x) = 2x/k(k+1) for x = 1, 2, 3, · · · , k

Answer 1

(a) $f(1)=-\frac{1}{5}, \;f(2)=0.\;f(3)=\frac{1}{5}.\;f(4)=\frac{2}{5},\;f(5)=\frac{3}{5}.$

All values of a probability mass function must be non-negative.

Thus, the given function cannot serve as the probability distribution.

(b) $f(1)=\frac{2}{k(k+1)},\;f(2)=\frac{4}{k(k+1)},\;...,f(k)=\frac{2k}{k(k+1)}.$

All values of f are positive.

$\Sigma_{i=1}^k=\frac{2}{k(k+1)}\Sigma_{i=1}^ki=\frac{2}{k(k+1)}*\frac{k(k+1)}{2}=1.$

Thus, the given function can serve as the probability distribution.