hw7 tags probability 2023PB_HW7.pdf 1.a 1=∫−∞∞ce−3x dx=ce−3x−3∣0∞limitx→∞1−3e3x=01=0−c1−3c=31=\int_{-\infty }^{\infty } ce^{-3x} \,dx={c\frac{e^{-3x}}{-3}}\Big|_{0}^{\infty}\\ \underset{x\rightarrow \infty}{\text{limit}}{\frac{1}{-3e^{3x}}}=0\\ 1=0-c\frac{1}{-3}\\ c=3\\1=∫−∞∞ce−3xdx=c−3e−3x0∞x→∞limit−3e3x1=01=0−c−31c=3 1.b ∫00.53e−3x dx=−e−3x∣00.5=−e−1.5+1=0.776\int_{0}^{0.5}3e^{-3x}\,dx ={{-e^{-3x}}}\Big|_{0}^{0.5}\\ =-e^{-1.5}+1=0.776∫00.53e−3xdx=−e−3x00.5=−e−1.5+1=0.776 2.a g(x)={0,x<432x−3,x≥4g(x)=\begin{cases} 0,x<4\\ 32x^{-3},x\ge 4\\ \end{cases}g(x)={0,x<432x−3,x≥4 2.b ∫040 dx+∫4532x−3 dx=−16x−2∣45=925∫6∞32x−3 dx=−16x−2∣6∞=49∫5732x−3 dx=−16x−2∣57=0.313∫13.50 dx=0\int_0^{4} 0\,dx+\int_4^5 32x^{-3}\,dx=-16x^{-2}\Big|_4^5=\frac{9}{25}\\ \int_6^{\infty} 32x^{-3}\,dx=-16x^{-2}\Big|_6^\infty=\frac{4}{9}\\ \int_5^{7} 32x^{-3}\,dx=-16x^{-2}\Big|_5^7=0.313\\ \int_1^{3.5} 0\,dx=0∫040dx+∫4532x−3dx=−16x−245=259∫6∞32x−3dx=−16x−26∞=94∫5732x−3dx=−16x−257=0.313∫13.50dx=0 3 y=x3p(x3≤y)=p(x≤y13)=∫−∞y1314 dx=14x∣−2y13=14y13+12g(y)={0,y<−814t13+12,−8≤y≤80,y>8g′(y)={0,y<−8112x−23,−8≤y≤80,y>8z=x4p(x4≤z)=p(z14≤x≤z14)=∫−z14z1414 dx=14x∣−z14z14=12z14g(z)={0,z<012z14,0≤z≤160,z>16g′(z)={0,z<0116z−34,0≤z≤160,z>16{\huge y=x^3}\\ p( x^3\le y)=p( x\le y^{\frac{1}{3}})=\int_{-\infty}^{y^{\frac{1}{3}}}\frac{1}{4}\,dx=\frac{1}{4}x\Big|^{y^{\frac{1}{3}}}_{-2}=\frac{1}{4}y^{\frac{1}{3}}+\frac{1}{2}\\ g(y)= \begin{cases} 0,y<-8\\ \frac{1}{4}t^{\frac{1}{3}}+\frac{1}{2},-8\le y \le 8\\ 0,y>8\\ \end{cases}\\ g'(y)=\begin{cases} 0,y<-8\\ \frac{1}{12}x^{-\frac{2}{3}},-8\le y \le 8\\ 0,y>8\\ \end{cases}\\ {\huge z=x^4}\\ p(x^4\le z)=p(z^{\frac{1}{4}}\le x\le z^{\frac{1}{4}})=\int_{-z^{\frac{1}{4}}}^{z^{\frac{1}{4}}}\frac{1}{4}\,dx=\frac{1}{4}x\Big|^{z^{\frac{1}{4}}}_{-z^{\frac{1}{4}}}=\frac{1}{2}z^{\frac{1}{4}}\\ g(z)= \begin{cases} 0,z<0\\ \frac{1}{2}z^{\frac{1}{4}},0\le z \le 16\\ 0,z>16\\ \end{cases}\\ g'(z)= \begin{cases} 0,z<0\\ \frac{1}{16}z^{\frac{-3}{4}},0\le z\le 16\\ 0,z>16\\ \end{cases}\\y=x3p(x3≤y)=p(x≤y31)=∫−∞y3141dx=41x−2y31=41y31+21g(y)=⎩⎨⎧0,y<−841t31+21,−8≤y≤80,y>8g′(y)=⎩⎨⎧0,y<−8121x−32,−8≤y≤80,y>8z=x4p(x4≤z)=p(z41≤x≤z41)=∫−z41z4141dx=41x−z41z41=21z41g(z)=⎩⎨⎧0,z<021z41,0≤z≤160,z>16g′(z)=⎩⎨⎧0,z<0161z4−3,0≤z≤160,z>16 4 y=x23p(0≤y≤t32)=∫0t32λe−λx dx=−eλx∣0t32=1−eλt32g(t)={0,t<01−e−λt32,0≤t<∞g′(t)={0,t<032λt12e−λt32,0≤t<∞y=x^{\frac{2}{3}}\\ p(0\le y\le t^{\frac{3}{2}})=\int_{0}^{t^{\frac{3}{2}}} \lambda e^{-\lambda x} \,dx=-e^{\lambda x}\Big|^{t^{\frac{3}{2}}}_{0}=1-e^{\lambda t^{\frac{3}{2}}}\\ g(t)= \begin{cases} 0,t<0\\ 1-e^{-\lambda t^{\frac{3}{2}}},0\le t < \infty \\ \end{cases}\\ g'(t)= \begin{cases} 0,t<0\\ \frac{3}{2}{\lambda}t^{\frac{1}{2}} e^{-\lambda t^{\frac{3}{2}}},0\le t < \infty \\ \end{cases}\\y=x32p(0≤y≤t23)=∫0t23λe−λxdx=−eλx0t23=1−eλt23g(t)={0,t<01−e−λt23,0≤t<∞g′(t)={0,t<023λt21e−λt23,0≤t<∞ 5 E(ex)=∫−∞∞ex×3e−3x dx=∫0∞3e−2x dx=−32e−2x∣0∞limx→∞−32e−2x=0E(ex)=0+32=32E(e^x)=\int_{-\infty}^{\infty} e^x\times 3e^{-3x}\, dx=\int_{0}^{\infty} 3e^{-2x}\, dx =\frac{-3}{2}e^{-2x}\Big|^\infty_0\\ \lim_{x\rightarrow\infty}\frac{-3}{2}e^{-2x}=0\\ E(e^x)=0+\frac{3}{2}=\frac{3}{2}E(ex)=∫−∞∞ex×3e−3xdx=∫0∞3e−2xdx=2−3e−2x0∞x→∞lim2−3e−2x=0E(ex)=0+23=23 6 E(x)=∫−∞∞x×12e−∣x∣ dx=∫0∞12xe−x dx+∫−∞012xex dx=−12(x+1)e−x∣0∞+12(x−1)ex∣−∞0=0+12−12+0=0E(x2)=∫−∞∞x2×12e−∣x∣ dx=2∫0∞12x2e−x dx=2Var(x)=E(x2)−E(x)2=2−02=2E(x)=\int_{-\infty}^{\infty} x\times \frac{1}{2} e^{-|x|}\, dx\\ =\int_{0}^{\infty} \frac{1}{2}x e^{-x}\, dx+\int_{-\infty}^{0} \frac{1}{2}x e^{x}\, dx\\ =-\frac{1}{2}(x+1) e^{-x}\Big|^\infty_0 +\frac{1}{2}(x-1) e^{x}\Big|_{-\infty}^0 \\ =0+\frac{1}{2}-\frac{1}{2}+0=0\\ E(x^2)=\int_{-\infty}^{\infty}x^2\times \frac{1}{2} e^{-|x|}\, dx\\ =2\int_{0}^{\infty}\frac{1}{2} x^2e^{-x}\, dx=2\\ Var(x)=E(x^2)-E(x)^2=2-0^2=2E(x)=∫−∞∞x×21e−∣x∣dx=∫0∞21xe−xdx+∫−∞021xexdx=−21(x+1)e−x0∞+21(x−1)ex−∞0=0+21−21+0=0E(x2)=∫−∞∞x2×21e−∣x∣dx=2∫0∞21x2e−xdx=2Var(x)=E(x2)−E(x)2=2−02=2