확률과 랜덤과정

Probability and Random Processes

0≤P (A) ≤1

P (S) = 1

사건 A의 확률 P (A)

P (A) = Underscript[lim, n→∞] n_A/n

P (Overscript[A, _]) = 1 - P (A)

P (φ) = 0

드모르간의 법칙

Union : Overscript[Overscript[Underscript[∪, i = 1], n] A_i, _] = Overscript[Underscript[∩, i = 1], n] Overscript[A_i, _]

Intersection : Overscript[Overscript[Underscript[∩, i = 1], n] A_i, _] = Overscript[Underscript[∪, i = 1], n] Overscript[A_i, _]

상호배타적 (mutually exclusive)

P (A_1∪A_2∪…∪A_n) = P (A_1) + P (A_2) + … + P (A_n)

A = A_1∪A_2∪…∪A_n

P (A) = P (A_1) + P (A_2) + … + P (A_n)

P (A∪B) = P (A) + P (B) - P (A∩B)

조건부 확률

P (A | B) = (P (A∩B))/(P (B))

베이즈 정리

{A_1, A_2, …, A_n} 상호배타적 (mutually exclusive)

P (A) = P (A_1) P (A | A_1) + P (A_2) P (A | A_2) + … + P (A_n) P (A | A_n)

= Underoverscript[∑, i = 1, arg3] P (A_i) P (A | A_i)

A = (A∩A_1) ∪ (A∩A_2) ∪…∪ (A∩A_n)

P (A) = P (A∩A_1) + P (A∩A_2) + … + P (A∩A_n)

P (A∩A_i) = P (A_i) P (A | A_i)

P (A) = P (A_1) P (A | A_1) + P (A_2) P (A | A_2) + … + P (A_n) P (A | A_n)

사건 A의 전체확률 (total probability)

베이즈 정리(Bayes' formula)

P (A_k | A) = (P (A_k∩A))/(P (A)) = (P (A_k∩A))/(Underoverscript[∑, i = 1, arg3] P (A_i) P (A | A_i))

독립사건

A, B - independence

P (A | B) = P (A)

P (A∩B) = p (A | B) P (B) = P (A) P (B)

P (A_i∩A_j) = P (A_i) P (A_j)

P (A_i∩A_j∩A_k) = P (A_i) P (A_j) P (A_k)

P (A_1∩A_2∩…∩A_n) = P (A_1) P (A_2) …P (A_n)

결합표본공간 (combined sample space, Cartesian Product Space)

S = S_1×S_2 = {(x_i, y_j) | x_i∈S_1, y_j∈S_2, i = 1, 2, …, N ; j = 1, 2, …, M}

S = S_1×S_2×…×S_N

순열 (Permutation)

n factorial

n× (n - 1) × (n - 2) ×…×3×2×1 = n !

0 != 1

P (n, r) = n !/(n - r) ! = n× (n - 1) ×…× (n - r + 1)

조합 (Combination)

C (n, r) = ({{n}, {r}}) = (P (n, r))/r ! = n !/((n - r) ! r !)

C (n, r) = C (n, n - r)

({{n + m}, {k}}) = Underoverscript[∑, i = 0, arg3] ({{n}, {i}}) ({{m}, {k - i}})

({{2n}, {n}}) = ({{n}, {0}}) ({{n}, {n}}) + ({{n}, {1}}) ({{n}, {n - 1}}) + … + ({{n}, {k}}) ({{n}, {n - k}}) + … + ({{n}, {n}}) ({{n}, {0}})

= ({{n}, {0}})^2 + ({{n}, {1}})^2 + ({{n}, {2}})^2 + … + ({{n}, {k}})^2 + … + ({{n}, {n}})^2

({{n}, {k}}) = ({{n}, {n - k}})

이항정리 (binomial theorem)

(a + b)^n = Underoverscript[∑, k = 0, arg3] ({{n}, {k}}) a^kb^(n - k)

Underoverscript[∑, k = 0, arg3] ({{n}, {k}}) = Underoverscript[∑, k = 0, arg3] ({{n}, {k}}) 1^k (1)^(n - k) = (1 + 1)^n = 2^n

스털링의 공식

n ! ~ (2π n)^(1/2) (n/)^n = (2π n)^(1/2) n^n^(-n)

신뢰성 함수 (reliability function)

R (t) = Underoverscript[∏, k = 1, arg3] R_k (t)

R (t) = 1 -[1 - R_1 (t)][1 - R_2 (t)] …[1 - R_n (t)] = 1 - Underoverscript[∏, k = 1, arg3][1 - R_k (t)]

랜덤변수

표본공간 S, w∈S

랜덤변수  X (w)

정의역 (domain) w, 치역 (range) X (w)

사건 A_x

A_x = {w | X (w) = x} =[X = x]

p = P[A_x]

[X≤x] = {w | X (w) ≤x}

[X>x] = {w | X (w) >x}

[a<X<x] = {w | a<X (w) <b}

분포함수

누적분포함수 (CDF: cumulative distribution function)

F_X (x) = P[X≤x] - ∞<x<∞

x_1<x_2, F_X (x_1) ≤F_X (x_2)

0≤F_X (x) ≤1

F_X (∞) = 1

F_X (-∞) = 0

P[a<X≤b] = F_X (b) - F_X (a)

P[X>a] = 1 - P[X≤a] = 1 - F_X (a)

이산랜덤변수 (discrete)

확률질량함수 (PMF: probability mass function) p_X(x)

p_X (x) = P[X = x]

F_X (x) = P[X≤x] = Underscript[∑, k≤x] p_X (k)

연속랜덤변수 (continuous)

P (X∈A) = ∫_Af_X (x) x

확률밀도함수 (PDF: probability density function)

f_X (x) = (F_X (x))/x

f_X (x) ≥0

∫_ (-∞)^∞f_X (x) x = 1

P[a≤X≤b] = ∫_a^bf_X (x) x

P[X = a] = ∫_a^af_X (x) x = 0

P[x<a] = P[x≤a] = F_X (a) = P[-∞≤X≤a] = ∫_ (-∞)^af_X (x) x

ex

∫_a^bcx =[cx] _a^b = c (b - a) = 1

c = 1/(b - a)

기대값 (Expectation)

데이터 집합

X_1, X_2, …, X_N

산술평균 (등차중앙)

Overscript[X, _] = (X_1 + X_2 + … + X_N)/N

가중 산술평균 가중치 w_1, w_2, …, w_N

Overscript[X, _] = (w_1X_1 + w_2X_2 + … + w_NX_N)/(w_1 + w_2 + … + w_N)

기대값

E[X] = Underscript[∑, i] x_ip_X (x_i)

E[X] = ∫_ (-∞)^∞x f_X (x) x

ex

p_K (k) = λ^k/k ! ^(-k)

E[K] = Underoverscript[∑, k = 0, arg3] k p_K (k)

= Underoverscript[∑, k = 0, arg3] k (λ^k/k ! ^(-k))

= Underoverscript[∑, k = 1, arg3] λ^k/(k - 1) ! ^(-k)

= λ^(-k) Underoverscript[∑, k = 1, arg3] λ^(k - 1)/(k - 1) !

= λ^(-k) Underoverscript[∑, k = 0, arg3] λ^k/k !

= λ^(-k) ^k

= λ

f_X (x) = λ ^(-λ x)

E[X] = ∫_ (-∞)^∞xf_X (x) x = ∫_0^∞x λ ^(-λ x) x

v = λ ^(-λ x) x

u = x

v = -^(-λ x)

u = x

E[X] =[u v] _0^∞ - ∫_0^∞vu

=[-x^(-λ x)] _0^∞ + ∫_0^∞^(-λ x) x

= 0 -[^(-λ x)/λ] _0^∞

= 1/λ

신뢰성 함수 R_X(t) : 음이 아닌 랜덤변수

R_X (t) = P[X>t] = 1 - P[X≤t] = 1 - F_X (t)

F_X (t) = 1 - R_X (t)

E[X] = ∫_0^∞P[X>x] x = ∫_0^∞[1 - F_X (x)] x

P[X>x] = ∫_x^∞f_X (u) u

{(x, u) | 0≤x<∞, x≤u<∞}

{(x, u) | 0≤x≤u, 0≤u<∞}

E[X] = Underoverscript[∑, k = 0, arg3] P[X>k]

E[X] = ∫_0^∞P[X>x] x - ∫_0^∞P[X< -x] x

f_X (x) = λ ^(-λ x)

x≥0

F_X (x) = ∫_0^xf_X (u) u = ∫_0^xλ ^(-λ u) u = 1 - ^(-λ x)

E[X] = ∫_0^∞P[X>x] x = ∫_0^∞[1 - F_X (x)] x

= ∫_0^∞^(-λ x) x = 1/λ

모멘트 (central moment)

E[x^m] = Overscript[X^m, _]

E[x^n] = Overscript[X^n, _] = Underscript[∑, i] x_i^np_X (x_i)

E[x^n] = Overscript[X^n, _] = ∫_ (-∞)^∞x^nf_X (x) x

E[(x - Overscript[X, _])^n] = Overscript[(X - Overscript[X, _])^n, _] = Underscript[∑, i] (x_i - Overscript[X, _])^np_X (x_i)

E[(x - Overscript[X, _])^n] = Overscript[(X - Overscript[X, _])^n, _] = ∫_ (-∞)^∞ (x - Overscript[X, _])^nf_X (x) x

n=2, 분산 (variance)

σ_X^2 = E[(x - Overscript[X, _])^2] = Overscript[(X - Overscript[X, _])^2, _] = Underscript[∑, i] (x_i - Overscript[X, _])^2p_X (x_i)

σ_X^2 = E[(x - Overscript[X, _])^2] = Overscript[(X - Overscript[X, _])^2, _] = ∫_ (-∞)^∞ (x - Overscript[X, _])^2f_X (x) x

E[Y] = E[aX + b] = aE[X] + b

σ_X^2 = E[(x - Overscript[X, _])^2] = E[X^2 - 2XOverscript[X, _] + (Overscript[X, _])^2]

= E[X^2] - 2E[X] Overscript[X, _] + (Overscript[X, _])^2

= E[X^2] - 2Overscript[X X, _] + (Overscript[X, _])^2

= E[X^2] - 2 (Overscript[X, _])^2 + (Overscript[X, _])^2

= E[X^2] - (Overscript[X, _])^2

= E[X^2] - (E[X])^2

σ_X^2 = E[X^2] - (E[X])^2

p_K (k) = λ^k/k ! ^(-k)

E[K] = Underoverscript[∑, k = 0, arg3] k p_K (k)

= Underoverscript[∑, k = 0, arg3] k (λ^k/k ! ^(-k))

= Underoverscript[∑, k = 1, arg3] λ^k/(k - 1) ! ^(-k)

= λ^(-k) Underoverscript[∑, k = 1, arg3] λ^(k - 1)/(k - 1) !

= λ^(-k) Underoverscript[∑, k = 0, arg3] λ^k/k !

= λ^(-k) ^k

= λ

E[K^2] = Underoverscript[∑, k = 0, arg3] k^2 (λ^k/k ! ^(-k)) = λ^(-λ) Underoverscript[∑, k = 1, arg3] (k λ^(k - 1))/(k - 1) !

Underoverscript[∑, k = 1, arg3] (k λ^(k - 1))/(k - 1) ! = /λ (λ^λ) = (1 + λ) ^λ

σ_K^2 = E[K^2] - (E[K])^2 = λ + λ^2 - λ^2 = λ

f_X (x) = λ ^(-λ x)

E[X^2] = ∫_ (-∞)^∞x^2f_X (x) x = ∫_0^∞x^2 λ ^(-λ x) x

v = λ ^(-λ x) x

u = x^2

v = -^(-λ x)

u = 2xx

E[X^2] =[u v] _0^∞ - ∫_0^∞vu

=[-x^2^(-λ x)] _0^∞ + ∫_0^∞2x ^(-λ x) x

= 0 + 2[-x^(-λ x)/λ] _0^∞ + ∫_0^∞ (2 ^(-λ x))/λx

= 2/λ^2

σ_K^2 = E[K^2] - (E[K])^2 = 2/λ^2 - 1/λ^2 = 1/λ^2

p_K(k)=(1 - p)^(k - 1)p

S = Underoverscript[∑, k = 0, arg3] a^k = aUnderoverscript[∑, k = 1, arg3] a^(k - 1) = a S

S = a S

S = 1/(1 - a)

Underoverscript[∑, k = 0, arg3] a^k = 1/(1 - a)

G = Underoverscript[∑, k = 0, arg3] (1 - p)^k

E[X] = -p G/p = -p/p (1/p) = -p(-1/p^2) = 1/p

조건부 기대값 (conditional expectation)

E[X | A] = Underscript[∑, i] x_ip_ (X | A) (x_i | A)

E[X | A] = ∫_ (-∞)^∞x f_ (X | A) (x | A) x

p_ (X | A) (x | A) = (p_X (x))/(P (A))

f_ (X | A) (x | A) = (f_X (x))/(P (A))

A = {X≤a}, -∞<a<∞

f_ (X | A) (x | X≤a) = (f_X (x))/P[X≤a] = (f_X (x))/(F_X (a))

E[X | A] = (∫_ (-∞)^ax f_X (x) x)/(F_X (a))

마르코프 부등식 (Markov inequality)

a>0

P[X≥a] ≤E[X]/a

E[X] = ∫_0^∞x f_X (x) x

= ∫_0^ax f_X (x) x + ∫_a^∞x f_X (x) x

≥ ∫_a^∞x f_X (x) x

≥ ∫_a^∞a f_X (x) x

= a∫_a^∞f_X (x) x

= a P[X≥a]

베르누이 시행 분포 (Bernoulli trial)

success, failure

0≤p≤1

P[success] = p

p[failure] = 1 - p

p_X (x) = p^x (1 - p)^(1 - x)

RowBox[{F_X (x), =, RowBox[{Cell[], {{0, x<0}, {1 - p, 0≤x<1}, {1, x≥1}}}]}]

E[X] = 0 (1 - p) + 1 (p) = p

E[X^2] = 0^2 (1 - p) + 1^2 (p) = p

σ_X^2 = E[X^2] - {E[X]}^2 = p - p^2 = p (1 - p)

이항분포

(a + b)^n = Underoverscript[∑, r = 0, arg3] C (n, r) a^(n - r) b^r = Underoverscript[∑, r = 0, arg3] C (n, r) a^rb^(n - r)

({{n}, {x}}) = n !/(x ! (n - x) !)

p_ (X (n)) (x) = ({{n}, {x}}) p^x (1 - p)^(n - x)

Underoverscript[∑, x = 0, arg3] p_ (X (n)) (x) = Underoverscript[∑, x = 0, arg3] ({{n}, {x}}) p^x (1 - p)^(n - x) = {p + (1 - p)}^n = 1^n = 1

E[X(n)]=n p

E[X (n)] = Underoverscript[∑, x = 0, arg3] x p_ (X (n)) (x)

= Underoverscript[∑, x = 0, arg3] x ({{n}, {x}}) p^x (1 - p)^(n - x)

= Underoverscript[∑, x = 0, arg3] xn !/(x ! (n - x) !) p^x (1 - p)^(n - x)

= Underoverscript[∑, x = 1, arg3] (x n (n - 1) !)/(x (x - 1) ! (n - x) !) p p^(x - 1) (1 - p)^(n - x)

= n p Underoverscript[∑, x = 1, arg3] (n - 1) !/((x - 1) ! (n - x) !) p^(x - 1) (1 - p)^(n - x)

j = x - 1

x = 1, j = 0

x = n, j = n - 1

E[X (n)] = n p Underoverscript[∑, j = 0, arg3] (n - 1) !/(j ! (n - 1 - j) !) p^j (1 - p)^(n - 1 - j)

= n p Underoverscript[∑, j = 0, arg3] ({{n - 1}, {j}}) p^j (1 - p)^(n - 1 - j)

= n p [p + (1 - p)]^(n - 1)

= n p (1)^(n - 1)

= n p

E[X^2(n)]=E[X(n){X(n)-1}]+E[X(n)]=n(n-1)p^2+np

E[X (n) {X (n) - 1}] = Underoverscript[∑, x = 0, arg3] x (x - 1) p_ (X (n)) (x)

= Underoverscript[∑, x = 0, arg3] x (x - 1) ({{n}, {x}}) p^x (1 - p)^(n - x)

= Underoverscript[∑, x = 0, arg3] (x (x - 1) n !)/(x ! (n - x) !) p^x (1 - p)^(n - x)

= Underoverscript[∑, x = 2, arg3] (x (x - 1) n (n - 1) (n - 2) !)/(x (x - 1) (x - 2) ! (n - x) !) p^2p^(x - 2) (1 - p)^(n - x)

= n (n - 1) p^2Underoverscript[∑, x = 2, arg3] (n - 2) !/((x - 2) ! (n - x) !) p^(x - 2) (1 - p)^(n - x)

j = x - 2

x = 2, j = 0

x = n, j = n - 2

E[X (n) {X (n) - 1}] = n (n - 1) p^2Underoverscript[∑, j = 0, arg3] (n - 2) !/(j ! (n - 2 - j) !) p^j (1 - p)^(n - 2 - j)

= n (n - 1) p^2Underoverscript[∑, j = 0, arg3] ({{n - 2}, {j}}) p^j (1 - p)^(n - 2 - j)

= n (n - 1) p^2 {p + (1 - p)}^(n - 2)

= n (n - 1) p^2

E[X^2 (n)] = E[X (n) {X (n) - 1}] + E[X (n)] = E[X^2 (n)] - E[X (n)] + E[X (n)] = n (n - 1) p^2 + np

σ_X(n)^2=n p(1-p)

σ_ (X (n))^2 = E[X^2 (n)] - {E[X (n)]}^2

= n (n - 1) p^2 + np - n^2p^2

= n p - n p^2

= n p (1 - p)

F_ (X (n)) (x)

F_ (X (n)) (x) = P[X (n) ≤x] = Underoverscript[∑, k = 0, arg3] ({{n}, {k}}) p^k (1 - p)^(n - k)

기하분포

p_X (x) = p (1 - p)^(x - 1)

E[X] = Underoverscript[∑, x = 1, arg3] x p_X (x) = Underoverscript[∑, x = 1, arg3] x p (1 - p)^(x - 1) = pUnderoverscript[∑, x = 1, arg3] x (1 - p)^(x - 1)

Underoverscript[∑, x = 0, arg3] (1 - p)^x = 1/p

/pUnderoverscript[∑, x = 0, arg3] (1 - p)^x = /p (1/p) = -1/p^2

Underoverscript[∑, x = 0, arg3] x (1 - p)^(x - 1) = 1/p^2

E[X] = pUnderoverscript[∑, x = 1, arg3] x (1 - p)^(x - 1) = p/p^2 = 1/p

E[X^2] = Underoverscript[∑, x = 1, arg3] x^2p_X (x) = (2 - p)/p^2

σ_X^2 = E[X^2] - (E[X])^2 = (1 - p)/p^2

P[X>n]

P[X>n] = Underoverscript[∑, x = n + 1, arg3] p (1 - p)^(x - 1)

= pUnderoverscript[∑, x = n + 1, arg3] (1 - p)^(x - 1)

= (1 - p)^npUnderoverscript[∑, x = n + 1, arg3] (1 - p)^(x - n - 1)

= (1 - p)^npUnderoverscript[∑, x = 0, arg3] (1 - p)^x

= ((1 - p)^np)/p

= (1 - p)^n

F_X(x)

F_X (x) = P[X≤x] = Underoverscript[∑, k = 1, arg3] p (1 - p)^(k - 1) = 1 - P[x>x] = 1 - (1 - p)^x

변형기하분포

p_Y (y) = (1 - p) P^(y - 1)

E[Y] = 1/(1 - p)

σ_Y^2 = p/(1 - p)^2

F_Y (y) = P[Y≤y] = Underoverscript[∑, k = 1, arg3] (1 - p) p^(k - 1) = 1 - P[X>y] = 1 - p^y

기하분포의 '건망증'특성 (forgetfulness property, memorylessness property)

K = X - n

k = 1, 2, …

p_ (K | X>n) (k | X>n) = P[K = k | X>n]

= P[X - n = k | X>n]

= P[X = n + k | X>n]

= P[(X = n + k) ∩ (X>n)]/p[X>n]

= P[X = n + k]/p[X>n]

= (p (1 - p)^(n + k - 1))/(1 - p)^n

= p (1 - p)^(k - 1)

= p_X (k)

파스칼(음의 이항)분포 (negative binomial distribution)

p_ (X (n - 1)) (k - 1) = ({{n - 1}, {k - 1}}) p^(k - 1) (1 - p)^(n - k)

X (n - 1) 은 (n - 1, p) 를 매개변수로 갖는 이항 랜덤변수 이다 .

p_X_k (n) = P[{X (n - 1) = k - 1} ∩ {X_1 = 1}] = P[{X (n - 1) = k - 1}] P[{X_1 = 1}]

= p_ (X (n - 1)) (k - 1) p_X_1 (1)

= ({{n - 1}, {k - 1}}) p^(k - 1) (1 - p)^(n - k) p

= ({{n - 1}, {k - 1}}) p^k (1 - p)^(n - k)

k = 1, 2, … ; n = k, k + 1, …

Underoverscript[∑, n = k, arg3] p_X_k (n) = Underoverscript[∑, n = k, arg3] ({{n - 1}, {k - 1}}) p^k (1 - p)^(n - k) = 1

q = 1 - p

E[X_k] = (1 - q)^kUnderoverscript[∑, n = k, arg3] n ({{n - 1}, {k - 1}}) q^(n - k)

음의 이항분포의 매클로린 급수 전개

g (q) = (1 - q)^(-k) = Underoverscript[∑, m = 0, arg3] ({{k + m - 1}, {k - 1}}) q^m = Underoverscript[∑, n = k, arg3] ({{n - 1}, {k - 1}}) q^(n - k)

/qg (q) = k/(1 - q)^(k + 1)

= Underoverscript[∑, n = k, arg3] (n - k) ({{n - 1}, {k - 1}}) q^(n - k - 1)

= 1/q {Underoverscript[∑, n = k, arg3] n ({{n - 1}, {k - 1}}) q^(n - k) - kUnderoverscript[∑, n = k, arg3] n ({{n - 1}, {k - 1}}) q^(n - k)}

k/(1 - q)^(k + 1) = 1/q {(E[X_k] - k)/(1 - q)^k}

E[X_k] = k/(1 - q) = k/p

E[X_k^2] = Underoverscript[∑, n = k, arg3] n^2p_X_k (n)

= Underoverscript[∑, n = k, arg3] n^2 ({{n - 1}, {k - 1}}) p^k (1 - p)^(n - k)

= p^kUnderoverscript[∑, n = k, arg3] n^2 ({{n - 1}, {k - 1}}) (1 - p)^(n - k)

^2/q^2g (q) = (k (k + 1))/(1 - q)^(k + 2) = Underoverscript[∑, n = k, arg3] (n - k) (n - k - 1) ({{n - 1}, {k - 1}}) q^(n - k - 2)

= 1/q^2Underoverscript[∑, n = k, arg3] (n^2 - 2n k - n + k^2 + k) ({{n - 1}, {k - 1}}) q^(n - k)

= 1/q^2 {(E[X_k^2] - (2k + 1) E[X_k] + k (k + 1))/(1 - q)^k}

E[X_k^2] = (k^2 + k q)/(1 - q)^2 = (k^2 + k (1 - p))/p^2

σ_X_k^2 = E[X_k^2] - (E[X_k])^2 = (k (1 - p))/p^2

F_X_k (x) = P[X_k≤x] = Underoverscript[∑, n = k, arg3] ({{n - 1}, {k - 1}}) p^k (1 - p)^(n - k)

초기하분포 (hypergeometric)

복원추출

N_1<N ⇒ A형태

N - N_1 ⇒ B형태

(n, p)

p = N_1/N

x = 0, 1, 2, …, n

p_X (x) = ({{n}, {x}}) (N_1/N)^x ((N - N_1)/N)^(n - x)

비복원추출

k = 0, 1, 2, …, min (n, N_1)

p_ (K_n (k)) = (({{N_1}, {k}}) ({{N - N_1}, {n - k}}))/({{N}, {n}})

E[K_n] = (n N_1)/N = n p

σ_K_n^2 = (n N_1 (N - N_1) (N - n))/(N^2 (N - 1)) = (n p (1 - p) (N - n))/(N - 1)

p = N_1/N, N→∞

E[K_n] = n p

σ_K_n^2 = n p (1 - p)

푸아송분포

p_K (k) = λ^k/k ! ^(-λ)

F_K (k) = P[K≤k] = Underoverscript[∑, r = 0, arg3] λ^r/r ! ^(-λ)

E[K] = Underoverscript[∑, k = 0, arg3] k p_K (k)

= Underoverscript[∑, k = 0, arg3] (k λ^k)/k ! ^(-λ)

= Underoverscript[∑, k = 1, arg3] λ^k/(k - 1) ! ^(-λ)

= λ ^(-λ) Underoverscript[∑, k = 1, arg3] λ^(k - 1)/(k - 1) !

= λ ^(-λ) Underoverscript[∑, k = 0, arg3] λ^k/k !

= λ ^(-λ) ^λ

= λ

Underoverscript[∑, k = 1, arg3] λ^k/(k - 1) ! = λUnderoverscript[∑, k = 1, arg3] λ^(k - 1)/(k - 1) ! = λ ^λ

σ_K^2 = E[K^2] - (E[K])^2 = λ^2 + λ - λ^2 = λ