hw8 1 2 3 4 5 6 7 8 9

hw8

tags probability

2023PB_HW8.pdf

1

p=14P(x)=Cx15px(1−p)15−xE(x)=∑i=015{i×P(x=i)}=3.75p=\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

2

x+y=kP=(x>13k)∩(y>13k)=(x>13k)∩(1−x>13k)=(13k<x<23)P=13x+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}\\

3

P(−ln(1−x)<y)=P(x<1−e−y)=∫01−e−y1dx=(1−e−y)g(y)={(1−e−y)0≤y≤∞0elseg′(y)={e−y0≤y≤∞0elseP(-ln(1-x)<y)=P(x<1-e^{-y})=\int_{0}^{1-e^{-y}}1dx=(1-e^{-y})\\ g(y)= \begin{cases} (1-e^{-y})& 0\le y \le \infty\\ 0 &else\\ \end{cases}\\ g'(y)= \begin{cases} e^{-y}&0\le y \le \infty\\ 0&else\\ \end{cases}\\

4

Φ(x)=12π∫−∞xe−t22dtP(x<Z<x+α)=Φ(x+α−μσ)−Φ(x−μσ)ddx(Φ(x+α−μσ)−Φ(x−μσ))=0(12πe−t2∣x−μσx+α−μσ)=0x=−12α\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}\Big|_{\frac{x-\mu}{\sigma}}^{\frac{x+\alpha-\mu}{\sigma}})=0\\ x=-\frac{1}{2}\alpha

5

μ=67,σ=64Φ(x)=12π∫−∞xe−t22dtP(x≥90)=1−P(x<90)=1−Φ(90−μσ)=0.00202P(80<x≤90)=Φ(90−μσ)−Φ(80−μσ)=0.050P(70<x≤80)=Φ(80−μσ)−Φ(70−μσ)=0.301P(60<x≤70)=Φ(70−μσ)−Φ(60−μσ)=0.4553P(x<60)=Φ(60−μσ)=0.1907\mu=67,\sigma=\sqrt{64}\\ \Phi (x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}{ e^{\frac{-t^2}{2}}dt}\\ P(x\ge90)=1- P(x<90)=1-\Phi(\frac{90-\mu}{\sigma})=0.00202\\ P(80<x\le90)=\Phi(\frac{90-\mu}{\sigma})-\Phi(\frac{80-\mu}{\sigma})=0.050\\ P(70<x\le80)=\Phi(\frac{80-\mu}{\sigma})-\Phi(\frac{70-\mu}{\sigma})=0.301\\ P(60<x\le70)=\Phi(\frac{70-\mu}{\sigma})-\Phi(\frac{60-\mu}{\sigma})=0.4553\\ P(x<60)=\Phi(\frac{60-\mu}{\sigma})=0.1907\\

6

P(∣x−μ∣>kσ)=P(x−μ>kσ)+P(−x+μ>kσ)=1−Φ(kσ+μ−μσ)+Φ(−kσ+μ−μσ)=1−Φ(k)+Φ(−k)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)

7

P(x<y)=P(x2<∣y∣)=P(−x2<y<x2)=Φ(x2)−Φ(−x2)=2Φ(x2)−1=2×12πe−(x2)22−1g(y)={22πe−y42−1y≥00elseg′(y)={−4y32πe−y42y≥00elseP(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{cases} \frac{2}{\sqrt{2\pi}}e^{\frac{-y^4}{2}}-1 &y\ge 0\\ 0 &else \end{cases}\\ g'(y)= \begin{cases} \frac{-4y^{3}}{\sqrt{2\pi}}e^{\frac{-y^{4}}{2}} &y\ge 0\\ 0 &else \end{cases}

8

P(x≤t)=1−e−λtE(x)=1λ,E(x2)=2λ2σ=E(x2)−E(x)2=1λP(∣x−E(x)∣>2σ)=P(x−E(x)>2σ)+P(x−E(x)<−2σ)=(1−1+e−λ(E(x)+σ))+(e−λ(E(x)−2σ))(−1λ<0)=(e−λ(2λ+1λ))=e−3P(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}

9

P(n>x+1n>x)=P(x>1)=1−P(x≤1)=1−λe−λp=1−λe−λg(x=i)=(1−λe−λ)(λe−λ)i−1=λe−λ−(λe−λ)iP(\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