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Distribusi Binomial Negatif

Distribusi binomial negatif adalah distribusi hasil percobaan bernoulli yang diulang sampai mendapatkan sukses ke-\(k\).

Fungsi Padat Peluang
\[ f(x)= \begin{cases} \displaystyle \binom{x-1}{k-1}p^k \left( 1-p\right)^{x-k} &\;\; x=k,k+1,k+2,... \\\\ 0 &\;\;\;\text{lainnya} \end{cases} \] Keterangan notasi:
\(p\) = peluang sukses
\(x\) = jumlah percobaan sampai mendapatkan sukses ke-\(k\)
\(k\) = jumlah sukses yang muncul

Mean

Mean dari distribusi binomial negatif adalah \( \displaystyle E(X) = \frac {k}{p}.\)
Bukti: \[ \begin{aligned} E(X)&= \sum_{x=k}^\infty xf(x) \\ &= \sum_{x=k}^\infty x \frac {(x-1)!}{(k-1)!(x-k)!}p^k \left( 1-p\right)^{x-k}\\ &= \sum_{x=k}^\infty \frac{kx!}{k!(x-k)!} \frac {p^{k+1}}{p}(1-p)^{x-k}\\ &= \frac {k}{p} \sum_{x=k}^\infty \frac{x!}{k!(x-k)!} p^{k+1} (1-p)^{x-k}\\ &= \frac {k}{p} \end{aligned} \]

Varian

Varian dati distribusi binomial negatif adalah \( \displaystyle Var(X)=\frac {k(1-p)}{p^2}.\)
Bukti: \[ \begin{aligned} Var(X)&=E\left (\left[X-E(X)\right]^2\right )\\ &=E(X^2)-\left [E(X) \right ]^2 \end{aligned} \] Untuk menyelesaikannya, tentukan bagian yang belum diketahui terlebih dahulu, yaitu \( E(X^2).\) \[ \begin{aligned} E(X^2) &= E(X^2)+E(X)-E(X)\\ &= E(X^2+X)-E(X)\\ &= E\left (X(X+1)\right )-E(X) \end{aligned}\] Selesaikan bagian \( E\left ( X(X+1) \right ).\) \[ \begin{aligned} E(X(X+1)) &= \sum_{x=k}^\infty x(x+1)f(x)\\ &= \sum_{x=k}^\infty x(x+1) \frac {(x-1)!}{(k-1)!(x-k)!}p^k \left( 1-p\right)^{x-k}\\ &= \sum_{x=k}^\infty \frac {k(k+1)(x+1)!}{(k+1)!(x-k)!} \frac {p^{k+2}}{p^2} \left(1-p\right)^{x-k}\\ &= \frac {k(k+1)}{p^2} \sum_{x=k}^\infty \frac {(x+1)!}{(k+1)!(x-k)!} p^{k+2} \left(1-p\right)^{x-k}\\ &= \frac {k(k+1)}{p^2}\end{aligned} \] Selanjutnya, \[ \begin{aligned} E(X^2) &= \frac {k(k+1)}{p^2} - \frac {k}{p}\\ &= \frac {k^2+k-kp}{p^2} \end{aligned} \] Dengan demikian, \[ \begin{aligned} Var(X) &= \frac {k^2+k-kp}{p^2} - \left (\frac {k}{p} \right )^2\\ &= \frac {k(1-p)}{p^2} \end{aligned} \]

Fungsi Pembangkit Momen (MGF)

Fungsi pembangkit momen distribusi binomial negatif adalah \( \displaystyle M_x(t)= \left ( \frac {pe^t}{1-(1-p)e^t} \right )^k\)
Bukti: \[ \begin{aligned} M_x(t) &= E(e^{tX}\\ &= \sum_{x=k}^\infty e^{tX} \frac {(x-1)!}{(k-1)!(x-k)!}p^k \left( 1-p \right )^{x-k}\\ &= \sum_{x=k}^\infty \frac {(x-1)!}{(k-1)!(x-k)!} \left ( pe^t \right )^k \left ( (1-p)e^t \right )^{x-k}\\ &= \left ( \frac {pe^t}{1-(1-p)e^t} \right )^k \sum_{x=k}^\infty \frac {(x-1)!}{(k-1)!(x-k)!} \left ( (1-p)e^t \right )^{x-k} \left ( 1-(1-p)e^t \right )^k\\ &= \left ( \frac {pe^t}{1-(1-p)e^t} \right )^k \end{aligned} \]

Fungsi Karakteristik



Fungsi Pembangkit Peluang



Baca juga:
  1. Nilai Harapan Distribusi Binomial Negatif