hw9 1.a 1.b 1.c 1.d 2 3.a 3.b 3.c 3.d 3.e 4 5 6 7.a 7.b 7.c

hw9

tags probability

2023PB_HW9.pdf

1.a

∑x∈1,2,3∑y∈1,2P(x,y)=1∑x∈1,2,3∑y∈1,2c(x+y)=121c=1c=121\sum_{x\in {1,2,3}} \sum_{y\in {1,2}} P(x,y)=1\\ \sum_{x\in {1,2,3}} \sum_{y\in {1,2}}c(x+y)=1\\ 21c=1\\ c=\frac{1}{21}

1.b

Px(x)=121(2x+3)Py(y)=121(3y+6)P_x(x)=\frac{1}{21}(2x+3)\\ P_y(y)=\frac{1}{21}(3y+6)

1.c

P(x≥2 ∣ y=1)=P(x≥2 ∩ y=1)Py(y=1)=1337=79P(x\ge2 \ | \ y=1)=\frac{P(x\ge 2 \ \cap \ y=1) }{P_y(y=1)}=\\ \frac{\frac{1}{3}}{\frac{3}{7}}=\frac{7}{9}

1.d

E(x)=∑x∈1,2,3x121(2x+3)=2.1904E(y)=∑2∈1,2y121(3y+6)=1.5714E(x)=\sum_{x\in {1,2,3}}x\frac{1}{21}(2x+3)=2.1904\\ E(y)=\sum_{2\in {1,2}}y\frac{1}{21}(3y+6)=1.5714

2

P(x,y)=x∖y11234521360000030236000041360236000502360236006136023602360702360236023681360236023609023602360010136023600011023600001213600000px(x)=136(6−∣7−x∣)py(y)={636y=0236(6−y)0<y≤50elseP(x,y)=\begin{array}{|c|c|c|c|c|c|c|} \hline x\setminus y &1&1 &2 &3 &4 &5 \\ \hline 2 &\frac{1}{36} &0&0&0&0&0\\ \hline 3 &0 &\frac{2}{36}&0&0&0&0\\ \hline 4 &\frac{1}{36} &0&\frac{2}{36}&0&0&0\\ \hline 5 &0 &\frac{2}{36}&0&\frac{2}{36}&0&0\\ \hline 6 &\frac{1}{36} &0&\frac{2}{36}&0&\frac{2}{36}&0\\ \hline 7 &0 &\frac{2}{36}&0&\frac{2}{36}&0&\frac{2}{36}\\ \hline 8 &\frac{1}{36} &0&\frac{2}{36}&0&\frac{2}{36}&0\\ \hline 9 &0 &\frac{2}{36}&0&\frac{2}{36}&0&0\\ \hline 10 &\frac{1}{36} &0&\frac{2}{36}&0&0&0\\ \hline 11 &0 &\frac{2}{36}&0&0&0&0\\ \hline 12 &\frac{1}{36} &0&0&0&0&0\\ \hline \end{array} \\\\ p_x(x)=\frac{1}{36}(6-|7-x|)\\ p_y(y)=\begin{cases} \frac{6}{36} & y=0\\ \frac{2}{36}(6-y) & 0<y\le 5\\ 0 & else \end{cases}

3.a

fx(x)=∫0x2 dy=2y∣0x=2xfy(y)=∫y12 dx=2x∣y1=2−2yf_{x}(x)={\int_0^x {2 } \ dy}=2y\Big|^x_0=2x \\ f_{y}(y)={\int_y^1 {2 } \ dx}=2x\Big|^1_y=2-2y \\

3.b

E(X)=∫01∫01x×fx(x) dydx=23E(Y)=∫01∫01y×fy(y) dydx=13E(X)=\int_0^1\int_0^1 x\times f_x(x)\ dydx=\frac{2}{3}\\ E(Y)=\int_0^1\int_0^1 y\times f_y(y)\ dydx=\frac{1}{3}\\

3.c

P(x<12)=∫0122x dx=14P(x<2y)=∫01∫v2v2 dudv=12P(x=y)=∫01∫vv2 dudv=0P(x<\frac{1}{2})=\int_0^\frac{1}{2}2x \ dx=\frac{1}{4}\\ P(x<2y)=\int_0^1\int_{\frac{v}{2}}^v 2 \ dudv=\frac{1}{2}\\ P(x=y)=\int_0^1\int_{v}^v 2 \ dudv=0\\

3.d

fxy≠fx(x)fy(y)not independentf_{xy}\neq f_x(x)f_y(y)\\ \text{not independent}

3.e

fx∣y(x∣y)=f(x,y)fy(y)=22−2yf_{x|y}(x|y)=\frac{f(x,y)}{f_y(y)}=\frac{2}{2-2y}

4

px(x)={37x=147x=20elsepy(y)={57y=127y=20elsepxy(1,1)=17px(1)py(1)=1549p(x,y)≠px(x)py(y)not independentp_x(x)= \begin{cases} \frac{3}{7} & x=1\\ \frac{4}{7} & x=2\\ 0 & else \end{cases}\\ p_y(y)= \begin{cases} \frac{5}{7} & y=1\\ \frac{2}{7} & y=2\\ 0 & else \end{cases}\\ p_{xy}(1,1)=\frac{1}{7}\\ p_x(1)p_y(1)=\frac{15}{49}\\ p(x,y)\neq p_x(x)p_y(y)\\ \text{not independent}

5

E(x2y)=∫0∞∫0∞x2y×2e−(x+2y) dxdy=1E(x^2y)=\int_0^\infty\int_0^\infty {x^2y\times 2e^{-(x+2y)}} \ dxdy=1

6

Px∣y(x∣y)=f(x,y)fy(y)Py(y)={525y=0725y=11325y=20elseP(x=2∣y=1)=fx,y(x=2,y=1)py(y=1)=525725=57E(x∣y=1)=∑xx×fx,y(x,y=1)=1225P_{x|y}(x|y)=\frac{f(x,y)}{f_y(y)}\\ P_y(y)=\begin{cases} \frac{5}{25}&y=0\\ \frac{7}{25}&y=1\\ \frac{13}{25}&y=2\\ 0& else\\ \end{cases}\\ P(x=2|y=1)=\frac{f_{x,y}(x=2,y=1)}{p_y(y=1)}=\frac{\frac{5}{25}}{\frac{7}{25}}=\frac{5}{7}\\ E(x|y=1)=\sum_{x} x\times{f_{x,y}(x,y=1)}=\frac{12}{25}\\

7.a

∫01∫01−xλ dydx=1λ=2\int_0^1\int_0^{1-x} \lambda \ dydx =1\\ \lambda=2

7.b

fx∣y(x∣y)=f(x,y)fy(y)fy(y)=∫01−y2 dx=2−2yfx∣y(x∣y)=11−yf_{x|y}(x|y)=\frac{f(x,y)}{f_y(y)}\\ f_y(y)=\int_0^{1-y}2 \ dx=2-2y\\ f_{x|y}(x|y)=\frac{1}{1-y}

7.c

E(X∣Y=y)=∫01−yx×fx∣y(x∣y) dx=∫01−yx×11−y dx=x22(1−y)∣01−y=1−y2E(X|Y=y)=\int_0^{1-y} x \times f_{x|y}(x|y) \ dx =\\ \int_0^{1-y} x \times \frac{1}{1-y} \ dx =\frac{x^2}{2(1-y)}\Big|_0^{1-y}=\frac{1-y}{2}\\