hw6 110590049

tags probability

2023PB_HW6.pdf

1

$$\begin{gathered} p=\frac{2}{12} \\ P(x=3)={\text{C}}_x^6\times p^x\times (1-p{)}^{6-x}=0.0535 \end{gathered}$$

2

$$\begin{gathered} p=0.45 \\ P(x)={\text{C}}_x^{10}\times p^x\times (1-p{)}^{10-x} \\ \text{when x=4 the P(x) have maximum probability }0.2383 \end{gathered}$$

3

$$\begin{gathered} p=0.999 \\ P(x)={\text{C}}_x^8\times p^{8-x}\times (1-p{)}^x \\ P(x=\{2,4,6,8\})=0.0000278324891609 \end{gathered}$$

4

$$\begin{gathered} p=0.025,E(x)=p\ast 80,\lambda =E(x) \\ P(x)=\frac{e^{-\lambda }\times {\lambda }^x}{x!} \\ P(X>=2)=1-P(0)-P(1)=0.59399415029 \end{gathered}$$

5

$$\begin{gathered} p=\frac{3}{10},E(x)=p\ast 35,\lambda =E(X) \\ P(x)=\frac{e^{-\lambda }\times {\lambda }^x}{x!} \\ P(x=10)=0.1236 \\ P(x=10{)}^2=0.015278 \end{gathered}$$

6

$$\begin{gathered} P(x)=\frac{e^{-\lambda }\times {\lambda }^x}{x!} \\ P(x=1)=\frac{e^{-\lambda }\times {\lambda }^1}{1!} \\ P(x=3)=\frac{e^{-\lambda }\times {\lambda }^3}{3!} \\ \frac{P(x=3)}{P(x=1)}=\frac{{\lambda }^2}{3!}=1 \\ \lambda =2.4494 \\ P(x=5)=0.0634449 \end{gathered}$$

7

$$\begin{gathered} p=0.2 \\ P(x,r)={\text{C}}_{r-1}^{x-1}\times (1-p{)}^{x-r}\times p^r \\ P(8,3)=0.05505 \end{gathered}$$

8

$$\begin{gathered} P(x)=p^{x-1}\ast (1-p) \\ P(x\ge n)=1-\sum _{i=1}^{n-1}P(i) \end{gathered}$$

9

$$\begin{gathered} p=0.15 \\ P(x)={\text{C}}_{10-1}^{x+10-1}\times p^{10}\times (1-p{)}^x \end{gathered}$$

10

$$\begin{gathered} p=\frac{50}{n},X\sim Hgeo(n,50,50) \\ d=50,t=50 \\ P(x)=\frac{{\text{C}}_x^d{\text{C}}_{t-x}^{n-d}}{{\text{C}}_t^n} \\ \frac{P_{n=624}(x=4)}{P_{n=625}(x=4)}\le 1,\frac{P_{n=626}(x=4)}{P_{n=625}(x=4)}\le 1 \\ n=625 \end{gathered}$$