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math-340:m340-f15-hw:hw-12 [2015/10/08 18:32]
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math-340:m340-f15-hw:hw-12 [2015/10/11 09:03] (current)
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   - $p_Z(z) = \begin{cases}\frac{1}{7},&​ \text{if } z = 0, \\ \frac{2}{7},&​ \text{if } z = 1, \\ \frac{2}{7},&​ \text{if } z = 4, \\ \frac{2}{7},&​ \text{if } z = 9, \\ 0,& \text{otherwise}.\end{cases}$   - $p_Z(z) = \begin{cases}\frac{1}{7},&​ \text{if } z = 0, \\ \frac{2}{7},&​ \text{if } z = 1, \\ \frac{2}{7},&​ \text{if } z = 4, \\ \frac{2}{7},&​ \text{if } z = 9, \\ 0,& \text{otherwise}.\end{cases}$
   - $p_T(t) = \begin{cases}\frac{1}{25},&​ \text{if } t = -4, 4, \\ \frac{2}{25},&​ \text{if } t = -3, 3, \\ \frac{3}{25},&​ \text{if } t = -2, 2, \\ \frac{4}{25},&​ \text{if } t = -1, 1, \\ \frac{1}{5},&​ \text{if } t = 0, \\ 0,& \text{otherwise}.\end{cases}$   - $p_T(t) = \begin{cases}\frac{1}{25},&​ \text{if } t = -4, 4, \\ \frac{2}{25},&​ \text{if } t = -3, 3, \\ \frac{3}{25},&​ \text{if } t = -2, 2, \\ \frac{4}{25},&​ \text{if } t = -1, 1, \\ \frac{1}{5},&​ \text{if } t = 0, \\ 0,& \text{otherwise}.\end{cases}$
-  -  +  - $p_S(s) = \begin{cases}\frac{1}{25},&​ \text{if } s = -4, 4, \\ \frac{2}{25},&​ \text{if } s = -3, 3, \\ \frac{3}{25},&​ \text{if } s = -2, 2, \\ \frac{4}{25},&​ \text{if } s = -1, 1, \\ \frac{1}{5},&​ \text{if } s = 0, \\ 0,& \text{otherwise}.\end{cases}$ 
-  - +  - $p_Z(z) = \begin{cases}\frac{2}{5},&​ \text{if } z = 0, \\ \frac{2}{5},&​ \text{if } z = 2, \\ \frac{1}{5},&​ \text{if } x = 6, \\ 0,& \text{otherwise}.\end{cases}$
   - $p_W(w) = \begin{cases}\frac{1}{100},&​ \text{if } w = 1, \\ \frac{3}{100},&​ \text{if } w = 2, \\ \frac{1}{20},&​ \text{if } w = 3, \\ \frac{7}{100},&​ \text{if } w = 4, \\ \frac{9}{100},&​ \text{if } w = 5,\\ \frac{11}{100},&​ \text{if } w = 6, \\ \frac{13}{100},&​ \text{if } w = 7, \\ \frac{3}{20},&​ \text{if } w = 8, \\ \frac{17}{100},&​ \text{if } w = 9, \\ \frac{19}{100},&​ \text{if } w = 10, \\ 0,& \text{otherwise}.\end{cases}$ $p_U(u) = \begin{cases}\frac{19}{100},&​ \text{if } w = 1, \\ \frac{17}{100},&​ \text{if } w = 2, \\ \frac{3}{20},&​ \text{if } w = 3, \\ \frac{13}{100},&​ \text{if } w = 4, \\ \frac{11}{100},&​ \text{if } w = 5,\\ \frac{9}{100},&​ \text{if } w = 6, \\ \frac{7}{100},&​ \text{if } w = 7, \\ \frac{1}{20},&​ \text{if } w = 8, \\ \frac{3}{100},&​ \text{if } w = 9, \\ \frac{1}{100},&​ \text{if } w = 10, \\ 0,& \text{otherwise}.\end{cases}$   - $p_W(w) = \begin{cases}\frac{1}{100},&​ \text{if } w = 1, \\ \frac{3}{100},&​ \text{if } w = 2, \\ \frac{1}{20},&​ \text{if } w = 3, \\ \frac{7}{100},&​ \text{if } w = 4, \\ \frac{9}{100},&​ \text{if } w = 5,\\ \frac{11}{100},&​ \text{if } w = 6, \\ \frac{13}{100},&​ \text{if } w = 7, \\ \frac{3}{20},&​ \text{if } w = 8, \\ \frac{17}{100},&​ \text{if } w = 9, \\ \frac{19}{100},&​ \text{if } w = 10, \\ 0,& \text{otherwise}.\end{cases}$ $p_U(u) = \begin{cases}\frac{19}{100},&​ \text{if } w = 1, \\ \frac{17}{100},&​ \text{if } w = 2, \\ \frac{3}{20},&​ \text{if } w = 3, \\ \frac{13}{100},&​ \text{if } w = 4, \\ \frac{11}{100},&​ \text{if } w = 5,\\ \frac{9}{100},&​ \text{if } w = 6, \\ \frac{7}{100},&​ \text{if } w = 7, \\ \frac{1}{20},&​ \text{if } w = 8, \\ \frac{3}{100},&​ \text{if } w = 9, \\ \frac{1}{100},&​ \text{if } w = 10, \\ 0,& \text{otherwise}.\end{cases}$
 +  - $P(U < W) = \frac{9}{10}$,​ $P(U \le W) = 1$; $U$ and $W$ are not independent:​ for example, $P(U = 10, W = 1) = 0 \ne p_U(10)p_W(1)$