====== 지수분포 (Exponential Distribution) ====
===== 표기 =====
 지수분포는 그 분포의 [[평균]]인 $\lambda^{-1}$를 이용해 표기한다.

  * $$ X \sim Exp(\lambda^{-1})$$

 단, $\theta \in ( \ 0 \ , \ \infty \ )$
===== 받침 =====
 $$ x \in ( \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ p(x) = \lambda e^{-\lambda x} $$

<plot>
 set title "Exponential Distribution PDF"
 set size 1
 set xrange [0:8]
 set yrange [0:2]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 plot (2)*exp(-x/0.5) title "Exp(0.5)", \
  (1)*exp(-x) title "Exp(1)", \
  (0.5)*exp(-x/2) title "Exp(2)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = 1 - e^{-\lambda x} $$

<plot>
 set title "Exponential Distribution CDF"
 set size 1
 set xrange [0:8]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 plot   1-exp(-x/0.5) title "Exp(0.5)", \
  1-exp(-x) title "Exp(1)", \
  1-exp(-x/2) title "Exp(2)"
</plot>
===== 기대값 =====
 $$ E(X) = \lambda^{-1} $$
===== 분산 =====
 $$ Var(X) = \lambda^{-2} $$
===== 왜도 =====
 $$ \gamma_{1} = 2 $$
===== 첨도 =====
 $$ \gamma_{2} = 6 $$
===== 특성함수 =====
 $$ \phi \ (t) = \left( 1-\frac{i t}{\lambda} \right) ^{-1} $$
===== 적률생성함수 =====
 $$ M(t) = \left( 1 - \frac{t}{\lambda} \right) ^{-1} $$
===== 원적률 =====
 $$ \mu'_{n} = \lambda^{-n} \cdot n! $$
===== 중심적률 =====
 $$ \mu_{n} =  \lambda^{-n} \cdot (n-1)! $$
===== 특성 =====
  - [[무기억성]]을 가진다.

----
  * [[분포]]
  * [[지수분포와 감마분포관계]]