코사인 제곱 확률질량함수의 구간 확률

$\cos ^{2} ( x ^\circ ) + \cos ^{2} ( 90 ^\circ - x ^\circ )$$= \cos ^{2} ( x ^\circ ) + \sin ^{2} ( x ^\circ )$$= 1$ 이고 $\displaystyle \cos ^{2} 45 = \frac{1}{2}$, $\cos ^{2} 90 ^\circ = 0$이므로 $\displaystyle \sum\limits _{x=1} ^{90} \cos ^{2} ( x ^\circ )$$\displaystyle = \sum\limits _{x=1} ^{44} \left\{ \cos ^{2} ( x ^\circ ) + \cos ^{2} ( 90 ^\circ - x ^\circ ) \right\}$$+ \cos ^{2} 45 ^\circ + \cos 90 ^\circ$ $\displaystyle = 44 \times 1 + \frac{1}{2}$$\displaystyle = \frac{89}{2}$ $\displaystyle \sum\limits _{x=1} ^{90} {\mathrm{P}} ( X = x )$$= 1$이므로 $\displaystyle \sum\limits _{x=1} ^{90} a \cos ^{2} ( x ^\circ )$$\displaystyle = \frac{89}{2} a = 1$ ∴ $\displaystyle a = \frac{2}{89}$ ∴ ${\mathrm{P}} ( 30 \leq {X} \leq 60 )$$\displaystyle = \frac{2}{89} \times \sum\limits _{x=30} ^{60} \cos ( x ^\circ )$ $\displaystyle = \frac{2}{89} \times \left[ \sum\limits _{x=30} ^{44} \left\{ \cos ^{2} ( x ^\circ ) + \cos ^{2} ( 90 ^\circ - x ^\circ ) \right\} + \cos ^{2} 45 ^\circ \right]$ $\displaystyle = \frac{2}{89} \left[ 15 + \frac{1}{2} \right]$ $\displaystyle = \frac{2}{89} \times \frac{31}{2}$$\displaystyle = \frac{31}{89}$