hw8 110590049

tags probability

2023PB_HW8.pdf

1

$$\begin{gathered} p=\frac{1}{4} \\ P(x)={\text{C}}_x^{15}{p}^x(1-p{)}^{15-x} \\ E(x)=\sum _{i=0}^{15}\{i\times P(x=i)\}=3.75 \end{gathered}$$

2

$$\begin{gathered} x+y=k \\ P=(x>\frac{1}{3}k)\cap (y>\frac{1}{3}k)= \\ (x>\frac{1}{3}k)\cap (1-x>\frac{1}{3}k)=(\frac{1}{3}k<x<\frac{2}{3}) \\ P=\frac{1}{3} \end{gathered}$$

3

$$\begin{gathered} P(-ln(1-x)<y)=P(x<1-e^{-y})={\int }_0^{1-e^{-y}}1dx=(1-e^{-y}) \\ g(y)=\{\begin{array}{ll}(1-e^{-y})& 0\le y\le \infty \\ 0& else\end{array} \\ {g}^{\prime }(y)=\{\begin{array}{ll}{e}^{-y}& 0\le y\le \infty \\ 0& else\end{array} \end{gathered}$$

4

$$\begin{gathered} \Phi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{\frac{-t^2}{2}}dt \\ P(x<Z<x+\alpha )=\Phi (\frac{x+\alpha -\mu }{\sigma })-\Phi (\frac{x-\mu }{\sigma }) \\ \frac{d}{dx}(\Phi (\frac{x+\alpha -\mu }{\sigma })-\Phi (\frac{x-\mu }{\sigma }))=0 \\ (\frac{1}{\sqrt{2\pi }}{e}^{-t^2}{|}_{\frac{x-\mu }{\sigma }}^{\frac{x+\alpha -\mu }{\sigma }})=0 \\ x=-\frac{1}{2}\alpha \end{gathered}$$

5

$$\begin{gathered} \mu =67,\sigma =\sqrt{64} \\ \Phi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{\frac{-t^2}{2}}dt \\ P(x\ge 90)=1-P(x<90)=1-\Phi (\frac{90-\mu }{\sigma })=0.00202 \\ P(80<x\le 90)=\Phi (\frac{90-\mu }{\sigma })-\Phi (\frac{80-\mu }{\sigma })=0.050 \\ P(70<x\le 80)=\Phi (\frac{80-\mu }{\sigma })-\Phi (\frac{70-\mu }{\sigma })=0.301 \\ P(60<x\le 70)=\Phi (\frac{70-\mu }{\sigma })-\Phi (\frac{60-\mu }{\sigma })=0.4553 \\ P(x<60)=\Phi (\frac{60-\mu }{\sigma })=0.1907 \end{gathered}$$

6

$$\begin{gathered} P(|x-\mu |>k\sigma )=P(x-\mu >k\sigma )+P(-x+\mu >k\sigma )= \\ 1-\Phi (\frac{k\sigma +\mu -\mu }{\sigma })+\Phi (\frac{-k\sigma +\mu -\mu }{\sigma })= \\ 1-\Phi (k)+\Phi (-k) \end{gathered}$$

7

$$\begin{gathered} P(x<y)=P(x^2<|y|)=P(-x^2<y<x^2)=\Phi (x^2)-\Phi (-x^2)=2\Phi (x^2)-1=2\times \frac{1}{\sqrt{2\pi }}{e}^{\frac{-(x^2{)}^2}{2}}-1 \\ g(y)=\{\begin{array}{ll}\frac{2}{\sqrt{2\pi }}{e}^{\frac{-y^4}{2}}-1& y\ge 0\\ 0& else\end{array} \\ {g}^{\prime }(y)=\{\begin{array}{ll}\frac{-4y^3}{\sqrt{2\pi }}{e}^{\frac{-y^4}{2}}& y\ge 0\\ 0& else\end{array} \end{gathered}$$

8

$$\begin{gathered} P(x\le t)=1-e^{-\lambda t} \\ E(x)=\frac{1}{\lambda },E(x^2)=\frac{2}{{\lambda }^2} \\ \sigma =\sqrt{E(x^2)-E(x{)}^2}=\frac{1}{\lambda } \\ P(|x-E(x)|>2\sigma )=P(x-E(x)>2\sigma )+P(x-E(x)<-2\sigma )= \\ (1-1+e^{-\lambda (E(x)+\sigma )})+(e^{-\lambda (E(x)-2\sigma )}{)}_{(-\frac{1}{\lambda }<0)}= \\ (e^{-\lambda (\frac{2}{\lambda }+\frac{1}{\lambda })})=e^{-3} \end{gathered}$$

9

$$\begin{gathered} P(\frac{n>x+1}{n>x})=P(x>1)= \\ 1-P(x\le 1)=1-\lambda e^{-\lambda } \\ p=1-\lambda e^{-\lambda } \\ g(x=i)=(1-\lambda e^{-\lambda })(\lambda e^{-\lambda }{)}^{i-1}= \\ \lambda e^{-\lambda }-(\lambda e^{-\lambda }{)}^i \end{gathered}$$