機率HW5
===

P1
---
X could be $0,1,2,3,4$ or $5$.
$P(X=0) = \frac{C^6_1}{C^6_1 C^6_1} = \frac{1}{6} = 0.1\bar{6}$
$P(X=1) = \frac{C^2_1+C^4_1\times 2}{C^6_1 C^6_1}=\frac{10}{36}=0.277777778$
$P(X=2) = \frac{C^4_1 + C^2_1 \times 2}{C^6_1 C^6_1} = \frac{8}{36}=0.222222222$
$P(X=3) = \frac{C^6_1}{C^6_1 C^6_1} = \frac{1}{6} = 0.1\bar{6}$
$P(X=4) = \frac{C^4_1}{C^6_1 C^6_1} = \frac{1}{9} = 0.\bar{1}$
$P(X=5) = \frac{C^2_1}{C^6_1 C^6_1} = \frac{1}{18} = 0.0\bar{5}$

$\Sigma^5_{i = 0}P(X=i) = 1$

P2
---
$P(X<1) = F(1 - \lim_{n \rightarrow 0^+}n) = \frac{1}{2}=0.5$
$P(X=1) =F(1) - F(1 - \lim_{n \rightarrow 0^+}n)=(\frac{1}{12}+\frac{7}{12}) -\frac{1}{2} = \frac{1}{6}=0.1\bar{6}$
$P(0 \le X < 1) = F(1 - \lim_{n \rightarrow 0^+}n) - F(0 - \lim_{n \rightarrow 0^+}n) = \frac{1}{2}-\frac{1}{4} = 1/4 = 0.25$
$P(X > \frac{1}{2}) = 1-F(\frac{1}{2}) = 1-\frac{1}{2}=0.5$
$P(X=\frac{3}{2}) = F(\frac{3}{2}) - F(\frac{3}{2} - \lim_{n \rightarrow 0^+}n)=0$
$P(1 < X \le 6) =F(6)-F(1) = 1-(\frac{1}{12}+\frac{7}{12})=\frac{1}{3}=0.\bar{3}$

P3
---
$0.88^i \le 0.4 \Rightarrow i \ge log_{0.88} 0.4 \Rightarrow i \ge 7.167852345999508$

### Ans: ++8++

P4
---
Define $p$ is p.m.f of X.
X could be $1,2,3,4,5$ or $6$.
$p(1) = P(X=1) = \frac{C^2_1C^6_1-1}{C^6_1 C^6_1} = \frac{11}{36} = 0.30\bar{5}$
$p(2) = P(X=2) = \frac{C^2_1C^5_1-1}{C^6_1 C^6_1} = \frac{9}{36} = 0.25$
$p(3) = P(X=3) = \frac{C^2_1C^4_1-1}{C^6_1 C^6_1} = \frac{7}{36} = 0.\bar{2}$
$p(4) = P(X=4) = \frac{C^2_1C^3_1-1}{C^6_1 C^6_1} = \frac{5}{36} = 0.19\bar{4}$
$p(5) = P(X=5) = \frac{C^2_1C^2_1-1}{C^6_1 C^6_1} = \frac{3}{36} = 0.08\bar{3}$
$p(6) = P(X=6) = \frac{C^2_1C^1_1-1}{C^6_1 C^6_1} = \frac{1}{36} = 0.02\bar{7}$

$\Sigma^6_{i = 1}p(i) = 1$

Define $F$ is distribution function of a random variable X.
$$
F(t)=
\left\{
\begin{eqnarray}
0 \qquad t<1\\
\Sigma^t_{i=1}\frac{12-(2i-1)}{36} \qquad t=1,2,3,4,5,6\\
1 \qquad t>6
\end{eqnarray}
\right.
$$

P5
---
$E(X) = 30\times \frac{4000}{2000000} + 800\times \frac{500}{2000000} + 1200000\times \frac{1}{2000000} -1 = -0.14$

### Ans: ++-0.14++

P6
---
$E[X(11-X)] = \Sigma^{10}_{x=1}x(11-x)\frac{1}{10} = \frac{1}{10}\Sigma^{10}_{x=1}(11x-x^2) =22$

### Ans: ++22++

P7
---
He should choose first business because it's standard deviation is less,means it is more stable.

### Ans: ++First business++

P8
---
Define $p$ is p.m.f.
$p(-3) = \frac{3}{8}$
$p(0) = \frac{3}{4}-\frac{3}{8}=\frac{3}{8}$
$p(6)=1-\frac{3}{4} = \frac{1}{4}$

$E(X) = (-3)\times p(-3) + 0 \times p(0) + 6 \times p(6) = (-3) \times \frac{3}{8} + 0 \times \frac{3}{8} + 6 \times \frac{1}{4} = \frac{3}{8}$
$E(X^2) = (-3)^2 \times p(3) + 0 \times p(0) + 6^2 \times p(6) = 9 \times \frac{3}{8} + 0 + 36 \times \frac{1}{4} = \frac{99}{8}$

$Var(X) = E(X^2) - E(X)^2 =\frac{99}{8} - \frac{9}{64} = \frac{783}{64}$
$\sigma = \sqrt{Var(X)} = \frac{3\sqrt{87}}{8}$

P9
---
$E[X(X-2)] = 3 = E[X^2-2X]=E[X^2]-2E[X]=E[X^2]-2 \Rightarrow E[x^2]=5$
$Var(X) = E(X^2) - E(X)^2 =5-1=4$
$Var(-3X+7) = 3^2 Var(X)=36$

### Ans: ++36++