확률질량함수와 g'(1)

이산확률변수 $X$의 확률질량함수가 $\displaystyle \mathrm{P} ( \mathit{X} = x ) = \frac{x}{15} ( x = 1 , 2 , 3 , 4 , 5 )$이므로 확률분포는 다음과 같다.

이 때, $\displaystyle \therefore \mathrm{E} ( X ) = \frac{1}{15} + \frac{4}{15} + \frac{9}{15} + \frac{16}{15} + \frac{25}{15} = \frac{11}{3}$ 한편, $\displaystyle g ( t ) = \sum\limits _{x = 1} ^{5} \mathrm{P} ( \mathit{X} = x ) \cdot t ^{x}$ $\displaystyle = \frac{1}{15} ( t + 2 t ^{2} + 3 t ^{3} + 4 t ^{4} + 5 t ^{5} )$ $\displaystyle g' ( t ) = \frac{1}{15} ( 1 + 4 t + 9 t ^{2} + 16 t ^{3} + 25 t ^{4} )$ $\displaystyle g' ( 1 ) = \frac{1}{15} ( 1 + 4 + 9 + 16 + 25 )$$\displaystyle = \frac{11}{3}$ $\therefore \mathrm{E} ( \mathit{2} X ) - g' ( 1 ) = 2 \mathrm{E} ( \mathit{X} ) - g' ( 1 )$$\displaystyle = \frac{11}{3}$