Pelemparan 1 Koin

Ruang sampel \(S = \{A, G\}\)

Variabel acak \(X\) = banyak Angka → \(X(A)=1,\ X(G)=0\)

Distribusi peluang:

\[ \begin{array}{c|cc} \hline x & 0 & 1 \\ \hline f(x) & \frac12 & \frac12 \\ \hline \end{array} \]

\(E(X)=\frac12,\quad Var(X)=\frac14\)

Pelemparan 2 Koin

Ruang sampel \(S = \{AA, AG, GA, GG\}\), masing-masing berpeluang \(\frac14\)

\(X\) = banyak Angka

\[ \begin{array}{c|c} \hline \text{Hasil} & X \\ \hline AA & 2 \\ AG & 1 \\ GA & 1 \\ GG & 0 \\ \hline \end{array} \] \[ \begin{array}{c|ccc} \hline x & 0 & 1 & 2 \\ \hline f(x) & \frac14 & \frac12 & \frac14 \\ \hline \end{array} \]

\(Y\) = selisih (Angka - Gambar)

\[ \begin{array}{c|c} \hline \text{Hasil} & Y \\ \hline AA & 2 \\ AG & 0 \\ GA & 0 \\ GG & -2 \\ \hline \end{array} \] \[ \begin{array}{c|ccc} \hline y & -2 & 0 & 2 \\ \hline f(y) & \frac14 & \frac12 & \frac14 \\ \hline \end{array} \]

\(Z\) = hasil kali (Angka × Gambar)

\[ \begin{array}{c|c} \hline \text{Hasil} & Z \\ \hline AA & 0 \\ AG & 1 \\ GA & 1 \\ GG & 0 \\ \hline \end{array} \] \[ \begin{array}{c|cc} \hline z & 0 & 1 \\ \hline f(z) & \frac12 & \frac12 \\ \hline \end{array} \]

\(E(X)=1,\quad E(Y)=0,\quad E(Z)=\frac12\)

Pelemparan 3 Koin

Ruang sampel \(S = \{AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG\}\), masing-masing berpeluang \(\frac18\)

\(Y\) = banyak Gambar

\[ \begin{array}{c|c} \hline \text{Hasil} & Y \\ \hline AAA & 0 \\ AAG & 1 \\ AGA & 1 \\ AGG & 2 \\ GAA & 1 \\ GAG & 2 \\ GGA & 2 \\ GGG & 3 \\ \hline \end{array} \] \[ \begin{array}{c|cccc} \hline y & 0 & 1 & 2 & 3 \\ \hline f(y) & \frac18 & \frac38 & \frac38 & \frac18 \\ \hline \end{array} \]

\(E(Y)=1,5,\quad Var(Y)=0,75\)

* \(Y\) = banyak Gambar, \(X\) = banyak Angka = \(3 - Y\)

1 Dadu

\(X\) = mata dadu

\[ \begin{array}{c|cccccc} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & \frac16 & \frac16 & \frac16 & \frac16 & \frac16 & \frac16 \\ \hline \end{array} \]

\(E(X)=3,5,\quad Var(X)=\frac{35}{12}\approx2,917\)

2 Dadu (Jumlah)

Distribusi peluang jumlah dua dadu:

\[ \begin{array}{c|ccccccccccc} \hline s & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline P(S=s) & \frac{1}{36} & \frac{2}{36} & \frac{3}{36} & \frac{4}{36} & \frac{5}{36} & \frac{6}{36} & \frac{5}{36} & \frac{4}{36} & \frac{3}{36} & \frac{2}{36} & \frac{1}{36} \\ \hline \end{array} \]

\(P(S=7)=\frac{6}{36}=\frac16,\quad E(S)=7,\quad Var(S)=\frac{35}{6}\)

+ 1 Dadu & 1 Koin

\(X\) = mata dadu (1-6), \(Y\) = hasil koin (A=1, G=0)

Distribusi peluang bersama \(f(x,y) = P(X=x, Y=y)\):

\[ \begin{array}{c|cc} \hline f(x,y) & y=0\ (G) & y=1\ (A) \\ \hline x=1 & \frac{1}{12} & \frac{1}{12} \\ x=2 & \frac{1}{12} & \frac{1}{12} \\ x=3 & \frac{1}{12} & \frac{1}{12} \\ x=4 & \frac{1}{12} & \frac{1}{12} \\ x=5 & \frac{1}{12} & \frac{1}{12} \\ x=6 & \frac{1}{12} & \frac{1}{12} \\ \hline \end{array} \]

\(P(X \ge 4, Y=1)=\frac{3}{12}=\frac14,\quad E(XY)=\frac{21}{12}=1,75\)