======Resultados de la Guía - Probabilidad y Estadística (61.06 y 61.09)======




=====Guía 1: Probabilidad=====
  - 
    - 0.56
    - 0.14
    - 0.04
    - 0.26
  - <tex>\mbox{Mujeres solteras no universitarias}=-57</tex>
  - 
    - 0.36
    - 0.64
    - 0.53
    - 0.17
    - 0.75
  - 
    - Roja
    - Roja
    - Blanca
  - 
    - 0.45
    - <tex>B=\{1,2,5\}</tex>
    - <tex>C=\{\{3\}\}</tex>
    - <tex>D=\{2\}</tex> o <tex>\{3\}</tex> o <tex>\{2,3\}</tex> o <tex>\{3,4\}</tex> o <tex>\{2,4\}</tex> o <tex>\{2,3,4\}</tex>
    - FIXME
  - 
    - 2/7
    - 16/35
    - 19/35
    - 2/5
  - El esposo.
  - 
    - 1/6
    - 13/18
    - 1/12
    - 1/4
    - 5/18
  - 
    - 0.5
    - <tex>\frac{\pi}{4}</tex>
    - <tex>P(C_{k})=\frac{\left(0.5+\frac{1}{k}\right)^{2}}{2}</tex> con <tex>k\geq2</tex>; <tex>P(\bigcap_{k=1}^{\infty}C_{k})=\frac{1}{8}</tex>
    - <tex>P(D_{k})=\pi\left(\frac{1}{k}\right)^{2}</tex>; <tex>P(D)=0</tex>
    - <tex>D_{k}</tex> representa el evento de que el dardo caiga dentro del círculo con centro en <tex>\left(\frac{1}{2},\frac{1}{2}\right)</tex> y radio <tex>\frac{1}{k}</tex>.\\ <tex>D</tex> representa el evento de que el dardo caiga en el centro del círculo. 
  - 
    - FIXME
    - <tex>P(\mbox{gane Juan})=P(\mbox{gane Maria})=\frac{5}{14}</tex>\\ <tex>P(\mbox{gane Pedro})=\frac{2}{7}</tex>\\ <tex>P(\mbox{nadie gana})=0</tex>
  - 
    - 1/1000
    - 1/59049
    - 0
  - 
    - Con reposición:
      - 1/64
      - 1/16
      - 1/1600
      - 3/8
    - Sin reposición:
      - 3/247
      - 12/247
      - 0
      - 100/247
  - 
    - 0.7015
    - 0.0489
  - 
    - 0.818
    - <tex>P(k)=\frac{{100-k \choose 10}{k \choose 0}}{{100 \choose 10}}+\frac{{100-k \choose 9}{k \choose 1}}{{100 \choose 10}}</tex> si <tex>0<k\leq90</tex>
    - 0.3329
  - FIXME
  - 0.916
  - Conviene apostar al resultado del evento "A"
  - 
    - 1/6
    - 1/6
    - 0
    - 1/2
    - 1/4
  - 
    - 0.56
    - 5/28
    - 8/55
    - 17/28
    - 7/32
    - 1/5
  - 
    - 60/143
    - 5/12
  - 
    - 12/248
    - 12/248
    - 0.04858
    - 158/63973
    - 0.0508
  - 
    - 1/3
    - 1/2
  - 
    - 3/4
    - 1/2
    - 3/8
    - 1/5
  - 
    - 0.44
    - 19/22
  - 9899/9900
  - 
    - 1/3
    - 40/63
    - 9/23
    - 17/21
    - Van a dar los mismos resultados porque las proporciones son las mismas.
  - FIXME
  - FIXME
  - 24/49
  - FIXME
  - 
    - 0.049
    - 0.00006734
  - 
    - 0.049
    - 0.000067
  - 
    - 0.0411
    - 0.086
    - FIXME
    - 0.0611
  - 
    - Al menos 5 respuestas correctas.
    - 0.3671
    - FIXME
  - 
    - 1/2
    - No son independientes.
  - FIXME
  - <tex>\begin{array}{|c|c|c|c|}\hline k & N=k & N\leq k & k\leq N\tabularnewline\hline \hline 0 & 400 & 400 & 900\tabularnewline\hline 1 & 400 & 800 & 500\tabularnewline\hline 2 & 100 & 900 & 100\tabularnewline\hline \end{array}</tex>
  - FIXME
  - 
    - 17/100
    - 1/150
    - 1/25
    - 23/25
    - <tex>\frac{\pi}{100}</tex>
  - 
    - <tex>\Omega=\{1,2,3,\ldots,\}</tex>
    - <tex>P\left(A_{1}\right)=\frac{1}{6}</tex>\\ <tex>P\left(A_{7}\right)=\left(\frac{5}{6}\right)^{6}\times\left(\frac{1}{6}\right)^{1}</tex>\\ <tex>P\left(A_{1017}\right)=\left(\frac{5}{6}\right)^{1016}\times\left(\frac{1}{6}\right)^{1}</tex>
    - <tex>P(B_{1})=1</tex>\\ <tex>P(B_{5})=0.4822</tex>\\ <tex>P(B_{1000})=0</tex>
    - FIXME
    - <tex>P(B)=0</tex>
  - 
    - <tex>P(\mbox{par})=0.422569</tex>
    - <tex>P(\mbox{dos pares})=0.47539</tex>
    - <tex>P(\mbox{terna})=0.021128</tex>
    - <tex>P(\mbox{escalera})=0.003940</tex>
    - <tex>P(\mbox{color})=0.001981</tex>
    - <tex>P(\mbox{full})=0.001441</tex>
    - <tex>P(\mbox{poker})=0.000240</tex>
    - <tex>P(\mbox{escalera real})=0.000015</tex>
  - FIXME
  - 
    - 1/2
    - 1/3
  - 
    * <tex>p_{0}=\frac{1}{4}</tex>
    * <tex>p_{8}=\frac{1}{4}</tex>
    * <tex>p_{5}=\frac{3}{8}</tex>
  - FIXME
  - 
    - 1/2
    - 1/2
    - <tex>\left(\frac{1}{2}\right)^{11}</tex>
    - 4/5
  - 
    - <tex>\frac{p}{p+q}</tex>
    - <tex>\frac{p}{1-r^{3}}</tex> siendo <tex>r=1-p-q</tex>
    - <tex>\frac{p}{p+q}</tex>


=====Guía 2: Funciones de densidad, distribución, conjuntas y marginales=====
  - <tex>F_{X}</tex> es monótona no decreciente\\ <tex>F_{X}</tex> es continua a derecha\\ <tex>\lim_{x\to-\infty}=0</tex>\\ <tex>\lim_{x\to\infty}=1</tex>
  - 
    - En <tex>X=\pm2</tex>, donde <tex>P(X=\pm2)=\frac{1}{3}</tex>
    - <tex>\frac{2}{3},1,\frac{2}{3},\frac{1}{3}</tex>
    - <tex>0,\frac{1}{3},0</tex>
    - <tex>1,\frac{2}{3}</tex>
    - <tex>\frac{1}{2}</tex>
  - 
    - <tex>F_{X}(0)=0</tex>
    - <tex>F_{X}(0.8)=0.1</tex>
    - <tex>F_{X}(2)=0.25</tex>
    - <tex>F_{X}\left(2\sqrt{2}\right)=0.5</tex>
    - <tex>F_{X}(4)=1</tex>
  - FIXME
  - 
    - <tex>F_{X}(x)=\begin{cases}0 & \mbox{si }0<x\\\frac{2}{3}x^{2}+\frac{1}{3}x & \mbox{si }0\leq x\leq1\\1 & \mbox{si }x>1\end{cases}</tex>
    - <tex>\frac{1}{3}</tex>, <tex>\frac{4}{9}</tex>
    - <tex>\frac{1}{10}</tex>
  - 
    - <tex>f_{T*}=\begin{cases}0 & \forall\mbox{ otro }t*\\e^{-t*} & \mbox{si }0\leq t*<1\end{cases}</tex>
    - <tex>e^{-1}</tex>
  - 
    - <tex>\begin{array}{|c|c|}\hline k & P(X=k)\tabularnewline\hline \hline 0 & 0.0609\tabularnewline\hline 1 & 0.2284\tabularnewline\hline 2 & 0.3427\tabularnewline\hline 3 & 0.2570\tabularnewline\hline 4 & 0.0963\tabularnewline\hline 5 & 0.0144\tabularnewline\hline \end{array}</tex>
    - <tex>\begin{array}{|c|c|}\hline k & P(X=k)\tabularnewline\hline \hline 1 & 0.1428\tabularnewline\hline 2 & 0.5714\tabularnewline\hline 3 & 0.2857\tabularnewline\hline \end{array}</tex>
  - FIXME
  - 
    - <tex>P(N=n)=\left(\frac{1}{4}\right)^{n-1}\left(\frac{3}{4}\right)</tex> para <tex>n\in N</tex>
    - 4/5
  - 
    - <tex>\theta\in\left[-\frac{1}{2},\frac{1}{2}\right]</tex>
    - <tex>\mbox{Mediana}=\frac{-1+\sqrt{1+4\theta^{2}}}{2\theta}</tex>
    - <tex>P(X<1)=1</tex>\\ <tex>P(0<X\leq1)=\frac{1+\theta}{2}</tex>\\ <tex>P\left(X<0|-\frac{1}{2}<X<\frac{1}{2}\right)=\frac{1}{2}-\frac{\theta}{4}</tex>
  - 3/4
  - 2/3
  - 
    - <tex>f_{X}(x)=\frac{3x(20-x)}{3514}</tex> para <tex>3\leq x\leq17</tex>
    - <tex>f_{X}(x)=\frac{x(20-x)}{162}</tex> para <tex>0<x<3</tex> o <tex>17<x<20</tex>
  - 
    - <tex>f_{X|X<3}(x)=\begin{cases}
0.5 & \mbox{si }0\leq x<2\\
2-\frac{2}{5}x & \mbox{si }2\leq x<3
\end{cases}</tex>
    - <tex>f_{X|X>3}(x)=\frac{5}{2}-\frac{x}{2}</tex> si <tex>3<x<5</tex>
  - 
    - <tex>f_{T}(t)=\beta\cdot\left(\frac{t}{\alpha}\right)^{\beta-1}\cdot e^{-\left(\frac{t}{\alpha}\right)^{\beta}}</tex>
    - 
      - Fallas casuales.
      - Fallas por desgaste.
      - Fallas tempranas.
      - Fallas tempranas.
      - Fallas por desgaste.
      - Fallas casuales.
    - FIXME
    - <tex>P(T>4|T>3)=1-e^{-\left(\frac{t}{\alpha}\right)^{\beta}}</tex>
  - 
    - Sin reposición\\ <tex>\begin{array}{|c||c|c|c|c}\cline{1-4} X\backslash Y & 0 & 1 & 2 & \mbox{Marginal de }X\tabularnewline\hline \hline 0 & 0 & 0.0285 & 0.0428 & \mathit{0.0714}\tabularnewline\hline 1 & 0.0428 & 0.2571 & 0.1285 & \mathit{0.4285}\tabularnewline\hline 2 & 0.1285 & 0.2571 & 0.0428 & \mathit{0.4285}\tabularnewline\hline 3 & 0.0428 & 0.0285 & 0 & \mathit{0.0714}\tabularnewline\hline \multicolumn{1}{c||}{\mbox{Marginal de }Y} & \mathit{0.2142} & \mathit{0.5714} & \mathit{0.2142} & \tabularnewline\end{array}</tex>\\ <tex>P(X+Y\leq2)=\frac{1}{2}</tex>
    - Con reposición\\ <tex>\begin{array}{|c||c|c|c|c|c|c}\cline{1-6} X\backslash Y & 0 & 1 & 2 & 3 & 4 & \mbox{Marginal de }X\tabularnewline\hline \hline 0 & 0.0197 & 0.0527 & 0.0527 & 0.0234 & 0.039 & 0.3164\tabularnewline\hline 1 & 0.0791 & 0.1582 & 0.1054 & 0.0234 & 0 & 0.4218\tabularnewline\hline 2 & 0.1186 & 0.1582 & 0.0527 & 0 & 0 & 0.2109\tabularnewline\hline 3 & 0.0791 & 0.0527 & 0 & 0 & 0 & 0.0468\tabularnewline\hline 4 & 0.0197 & 0 & 0 & 0 & 0 & 0.0039\tabularnewline\hline \multicolumn{1}{c|}{\mbox{Marginal de }Y} & 0.3164 & 0.4218 & 0.2142 & 0.0468 & 0.0039 & \tabularnewline\end{array}</tex>\\ <tex>P(X+Y\leq2)=0.4812</tex>
  - 
    - 1/2
    - <tex>f_{X}(x)=\left(\frac{2}{\pi}\right)\sqrt{1-x^{2}}</tex> para <tex>x\in[-1,1]</tex>\\ <tex>f_{Y}(y)=\left(\frac{4}{\pi}\right)\sqrt{1-y^{2}}</tex> para <tex>y\in[0,1]</tex>
    - No son independientes.
  - 
    - 5/12
    - <tex>f_{X}(x)=4x(1-x^{2})</tex> para <tex>0\leq x\leq1</tex>\\ <tex>f_{Y}(y)=4y^{3}</tex> para <tex>0\leq y\le1</tex>
    - No son independientes.
  - 8/9
  - 
    - 0.1035
    - 4/9
    - 0.127
  - 
    - 1/50
    - 47/100
    - 17/18
    - <tex>f_{T}(t)=\begin{cases}\frac{t-22}{136} & \mbox{ si }22<t<34\\\frac{12}{136} & \mbox{ si }34<t<36\\\frac{48-t}{136} & \mbox{ si }36<t<40\end{cases}</tex>


=====Guía 3: Esperanza, varianza y covarianza=====
  - 
    - 1.1
    - <tex>p_{3}=0.6</tex>\\ <tex>x_{3}=0</tex>
    - 1.625
    - 2.2
  - 
    - 2/3
    - 1
    - 7/9
    - 7.2
  - <tex>\frac{2}{3}\theta</tex>
  - 
    - 19/18
    - <tex>E[X|X<1]=\frac{2}{3}</tex>\\ <tex>E[X|X\leq1]=\frac{5}{6}</tex>
    - <tex>f_{X|X>1}(x)=1</tex> para <tex>1<x<2</tex>
  - 
    - <tex>p_{1}=p_{3}=0.5</tex>\\ <tex>p_{2}=0</tex>
    - <tex>p_{1}=p_{3}=0</tex>\\ <tex>p_{2}=1</tex>
  - 
    - <tex>\frac{1800}{\pi}</tex>
    - <tex>\sqrt{15}\pi</tex>
  - 
    - <tex>\mu_{y}=2</tex>\\ <tex>\sigma_{y}^{2}=36</tex>
    - <tex>\mu_{y}=27</tex>
    - <tex>\mu_{y}=16</tex>
    - <tex>c=2</tex>
    - <tex>a=\pm\frac{1}{3}</tex>, <tex>b=\mp\frac{2}{3}</tex>
    - <tex>-11.4<a<11.4</tex>
  - <tex>-\frac{1}{2}</tex>
  - 
    - <tex>E[N]=\frac{19}{9}</tex>\\ <tex>var(N)=0.3209</tex>\\ <tex>cov(N,X_{1})=0</tex>
    - FIXME
    - <tex>cov(X_{i},X_{i})=var(X_{i})=\frac{2}{3}</tex>, <tex>cov(X_{i},X_{j})=-\frac{1}{3}</tex>
  - <tex>\rho=-0.123</tex>
  - FIXME
  - 
    - 1/9
    - 2/3
    - 2/3
  - 
    - <tex>E[Z]=\frac{2}{3}</tex>,\\ <tex>var(Z)=\frac{1}{18}</tex>\\ <tex>E[W]=\frac{1}{3}</tex>\\ <tex>var(W)=\frac{1}{18}</tex>
    - 7/18
    - 1/36
  - 
    - <tex>E[S_{n}]=2n</tex>\\ <tex>var(S_{n})=9n</tex>
    - FIXME
    - <tex>n>9000</tex>


=====Guía 4: Transformación de variables aleatorias=====
  - 
    - <tex>f_{Y}(y)=\frac{1}{9}\left(\frac{y-b}{a}+1\right)^{2}</tex> si <tex>b-a<y<2a+b</tex>
    - <tex>f_{Y}(y)=\frac{\left(\sqrt[3]{-y}+1\right)^{2}}{27(-y)^{2/3}}</tex> si <tex>-8<y<0</tex> y <tex>0<y<1</tex>
    - <tex>f_{Y}(y)=\frac{\left(\frac{3}{2}-\sqrt{9+4y}\right)^{2}+\left(\frac{3}{2}+\sqrt{9+4y}\right)^{2}}{18\sqrt{9+4y}}\mbox{ si }0<y<1,25</tex>
    - <tex>f_{Y}(y)=\begin{cases}\frac{\left(1+\sqrt{y}\right)^{2}+\left(1-\sqrt{y}\right)^{2}}{18\sqrt{y}} & \mbox{si }0<y<1\\\frac{\left(1+\sqrt{y}\right)^{2}}{18\sqrt{y}} & \mbox{si }1<y<4\end{cases}</tex> 
    - <tex>F_{Y}(y)=\begin{cases}\frac{1}{27}\left(y+1\right)^{2} & \mbox{si }-1<y<1\\1 & \mbox{si }y\geq1\end{cases}</tex>
  - <tex>f_{X}(x)=\frac{1}{\pi(1+x^{2})}</tex> para todo <tex>x</tex>
  - FIXME
  - <tex>f_{L}(l)=\frac{l}{30\pi}e^{-\frac{l^{2}}{60\pi}}</tex> si <tex>l>0</tex>
  - <tex>y=\begin{cases}\frac{1-e^{-x}}{3} & \mbox{si }0<x<\ln(4)\\1-3e^{-x} & \mbox{si }x>\ln(4)\end{cases}</tex>
  - <tex>f_{T}(t)=\begin{cases}\frac{1}{10} & \mbox{si }0<t<5\\\frac{1}{20} & \mbox{si }5\leq t<15\end{cases}</tex>
  - <tex>F_{V_{2}}(v_{2})=\begin{cases}\frac{1}{2}v_{2}+\frac{1}{4} & \mbox{si }0\leq v_{2}<1\\1 & \mbox{si }v_{2}\geq1\end{cases}</tex>
  - <tex>P(N=k)=e^{-\lambda k}(e^{\lambda}-1)</tex> para <tex>k\in N</tex>
  - <tex>f_{Z}(z)=\begin{cases}z+1 & \mbox{si }-1\leq z<0\\1-z & \mbox{si }0\leq z<1\end{cases}</tex>
  - 
    - <tex>f_{X}(x)=1</tex> si <tex>0<x<1</tex>\\ <tex>f_{Y}(y)=1</tex> si <tex>0<y<1</tex>
    - <tex>Z=X+Y</tex>\\ <tex>f_{Z}(z)=\begin{cases}z & \mbox{si }0<z<1\\2-z & \mbox{si }1<z<2\end{cases}</tex>
  - <tex>f_{Z}(z)=\begin{cases}\frac{45}{4}z^{2} & \mbox{si }0<z<0.5\\\frac{3}{4z^{2}}-\frac{3}{4}z^{2} & \mbox{si }0.5<z<1\end{cases}</tex>
  - FIXME
  - <tex>f_{U}(u)=\begin{cases}1.6(1-u) & \mbox{si }0<u<0.5\\3.2(1-u) & \mbox{si }0.5<u<1\end{cases}</tex>
  - 
    - <tex>f_{U}(u)=\left(\lambda_{1}+\lambda_{2}\right)e^{-u\left(\lambda_{1}+\lambda_{2}\right)}</tex> para <tex>u>0</tex>
    - <tex>P(j=1)=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}</tex>\\ <tex>P(j=2)=\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}}</tex>
  - FIXME
  - 0.2314
  - <tex>f_{Z|Z<\frac{1}{4}}(z)=4</tex> para <tex>0<z<\frac{1}{4}</tex>
  - 
    - <tex>f_{Z}(z)=\begin{cases}2 & \mbox{si }0<z<\frac{1}{4}\\\frac{2\left(1-\sqrt{z}\right)}{\sqrt{z}} & \mbox{si }\frac{1}{4}<z<1\end{cases}</tex>
    - <tex>f_{V}(v)=\begin{cases}-\frac{148}{9}v+\frac{14}{3} & \mbox{si }0<v<0.25\\-\frac{4}{9}v+\frac{2}{3} & \mbox{si }0.25<v<1.5\end{cases}</tex>
  - FIXME
  - FIXME
  - <tex>f(u,v)=u\lambda^{2}e^{-\lambda u}</tex> para <tex>u>0</tex> y <tex>0<v<1</tex>\\ <tex>f(u)=u\lambda^{2}e^{-\lambda u}</tex> para <tex>u>0</tex>\\ <tex>f(v)=1</tex> para <tex>0<v<1</tex>\\ Son independientes.
  - FIXME
  - FIXME
  - <tex>f_{X}(x)=\frac{1}{\pi(x^{2}+1)}</tex> si <tex>x\in(-\infty,\infty)</tex>
  - FIXME
  - 
    - 0.0128\\ 0.08
    - <tex>F_{Y}(y)=\begin{cases}0 & \mbox{ si }y<0\\0.0128 & \mbox{ si }y=0\\\frac{2}{625}y^{2}+\frac{8}{625}y+0.0128 & \mbox{ si }0<y<10.5\\-\frac{2}{625}y^{2}+\frac{92}{625}y-0.6928 & \mbox{ si }10.5<y<18\\1 & \mbox{ si }y\geq18\end{cases}</tex>


=====Guía 5: Esperanza y varianza condicional, suma de variables aleatorias, mezcla, truncamiento=====
  - 
    - <tex>f_{Y|X=0.75}(y)=\frac{4}{3}</tex> si <tex>0<y<0.75</tex>\\ <tex>f_{Y|X=0.4}(y)=\frac{5}{2}</tex> si <tex>0<y<0.4</tex>
    - No son independientes.
    - <tex>\frac{1}{3}</tex>, 0,375
  - 
    - <tex>f_{Y|X=x}(y)=\begin{cases}1 & \mbox{ si }0<y<1;0<x<0.5\\\frac{2}{3} & \mbox{ si }0<y<0.5;0.5<x<1\\\frac{4}{3} & \mbox{ si }0.5<y<1;0.5<x<1\end{cases}</tex>
    - 0.5
  - 
    - <tex>f(y_{1},y_{2})=\lambda^{2}y_{1}e^{-\lambda y_{1}}\frac{1}{y_{2}^{2}+2y_{2}+1}\mbox{ si }y_{1}>0;y{}_{2}>0</tex>
    - <tex>f(y_{1},y_{3})=\frac{\lambda^{2}}{2}e^{-\lambda y_{1}}\begin{array}{ccc} {} & {} & {}\end{array}y_{1}>0;y_{1}>y{}_{3};y_{1}>-y{}_{3}</tex>
    - <tex>\begin{array}{l}f_{y_{1}|y_{2}=1}(y_{1})=f(y_{1})=\lambda^{2}y_{1}e^{-\lambda y_{1}}\mbox{ si }y_{1}>0\\f_{y1|y3=0}(y_{1})=\lambda e^{-\lambda y_{1}}\mbox{ si }y_{1}>0\end{array}</tex>
    - <tex>A=2e^{-1}</tex>
    - <tex>B=e^{-1}</tex>
  - FIXME
  - <tex>f(Cc3)=\begin{cases}0.2689x & \mbox{ si }0<x<2\\0.2689(4-x) & \mbox{ si }2<x<3\\0.1176(4-x) & \mbox{ si }3<x<4\end{cases}</tex>
  - 
    - 0,439
    - 0,851
  - <tex>\begin{array}{l}f(x)=\frac{1}{2\sqrt{2\pi}\sigma}\left(e^{-\frac{1}{2}\left(\frac{x+1}{\sigma}\right)^{2}}+e^{-\frac{1}{2}\left(\frac{x-1}{\sigma}\right)^{2}}\right)\\\psi(x)=\frac{e^{-x/\sigma^{2}}}{e^{-x/\sigma^{2}}+e^{x/\sigma^{2}}}\end{array}</tex>
  - 
    - <tex>F_{Y|X=x}(y)=1-e^{xy}</tex> si <tex>x>0</tex>, <tex>y>0</tex>
    - <tex>\varphi(x)=\frac{1}{x}</tex> si <tex>x>0</tex>
    - <tex>E[Y|X]=\frac{1}{X}</tex>
    - <tex>F(z)=e^{-\frac{1}{z}}</tex> si <tex>z>0</tex>
    - 0,5811
  - 
    - <tex>F{}_{Y|X=x}(y)=\frac{y}{x}</tex> si <tex>0<y<x</tex>, <tex>0<x<1</tex>
    - <tex>\varphi(x)=\frac{x}{2}</tex> si <tex>0<x<1</tex>\\ <tex>\varphi(x)=\frac{x^{2}}{12}</tex> si <tex>0<x<1</tex>
    - <tex>E[Y|X]=\frac{X}{2}</tex>\\ <tex>E[Y|X]=\frac{X^{2}}{12}</tex>
    - <tex>\frac{1}{4}</tex>, <tex>\frac{9}{16}</tex>
  - <tex>Z=E[Y|X]</tex>\\ <tex>W=var[Y|X]</tex>Z\\ <tex>P\left(Z=\frac{3}{8}\right)=\frac{9}{10}</tex>\\ <tex>P\left(Z=\frac{7}{8}\right)=\frac{1}{10}</tex>\\ <tex>P\left(W=\frac{3}{64}\right)=\frac{9}{10}</tex>\\ <tex>P\left(W=\frac{1}{192}\right)=\frac{1}{10}</tex>\\ <tex>P\left(W<\frac{1}{32}\right)=\frac{1}{10}</tex>
  - 
    - <tex>P_{Y|X=x}(y)=\frac{{4 \choose y}{2 \choose 3-x-y}}{{6 \choose 3-x}}\mbox{ si }x=0,1,2,3;\,0\le y\le3-x</tex>.
    - <tex>E[Y|X=x]=(3-x)\frac{4}{6}</tex>\\ <tex>var[Y|X=x]=\left(9-x^{2}\right)\frac{2}{45}</tex>
    - <tex>E[Y|X]=(3-X)\frac{4}{6}</tex>\\ <tex>V[Y|X]=\left(9-X^{2}\right)\frac{2}{45}</tex>
  - <tex>\varphi(x)=E[Y|X]=\frac{e^{x/\sigma^{2}}-e^{-x/\sigma^{2}}}{e^{-x/\sigma^{2}}+e^{x/\sigma^{2}}}</tex>
  - 
    - <tex>E[Y|X]=0</tex>\\ Son independientes.
    - <tex>E[Y|X]=0</tex>\\ No son independientes.
  - 14
  - 0
  - <tex>Y=100-X</tex>\\ <tex>cov(X,Y)=-25</tex>
  - 2
  - 5 mm a la derecha.
  - 1
  - <tex>E[Y]=7.5</tex>\\ <tex>var(Y)=5.75</tex>
  - <tex>E[T]=1.266</tex>\\ <tex>var(T)=0.8434</tex>
  - 
    - <tex>E[Y|X]=\frac{X^{2}}{2}</tex> si <tex>0<x<2</tex>
    - <tex>Y=4X-\frac{14}{5}</tex>
  - 
    - <tex>E[Y|X]=X^{2}</tex>
    - <tex>Y=20X-50</tex>
  - FIXME


=====Guía 6: Procesos Bernoulli, geométrico e hipergeométrico=====
  - 
    - 0,6241
    - 0,6992
  - 
    - 0,6651
    - 0,4018
    - 0,2009
  - 
    - 1 y 2
    - 2
    - <tex>\frac{n}{5}</tex>
  - 0,8233
  - 0.05174
  - <tex>n\geq22</tex>
  - 
    - 0.0531
    - 0.2373
  - <tex>E[N]=11.4</tex>\\ <tex>var(N)=25.17</tex>
  - 40
  - 0,03348
  - 0,1273
  - FIXME
  - 
    - 0,00277
    - 0,1527
    - 0,5042
  - <tex>P(r=1)=\begin{cases}0 & \mbox{si }k=0\\\frac{1}{3} & \mbox{si }k=1\\\frac{10}{21} & \mbox{si }k=2\\\frac{15}{28} & \mbox{si }k=3\\\frac{5}{9} & \mbox{si }k=4\\\frac{5}{9} & \mbox{si }k=5\\\frac{6}{81} & \mbox{si }k=6\end{cases}</tex>\\ Es más probable cuando <tex>k=4</tex> o <tex>k=5</tex>.
  - 
    - <tex>P(Y=y)=\begin{cases}\frac{1}{6} & \mbox{si }y=0\\\frac{2}{3} & \mbox{si y=1}\\\frac{1}{6} & \mbox{si }y=2\end{cases}</tex>
    - <tex>P(X=x)=\begin{cases}\frac{217}{270} & \mbox{si }x=0\\\frac{26}{135} & \mbox{si }x=1\\\frac{1}{270} & \mbox{si }x=2\end{cases}</tex>
    - 19,63%
  - FIXME
  - 
    - 0,1297
    - 0,3589
    - 0,9896
    - 0,2969
    - 0,9999
    - 0,00065
  - 0,303
  - FIXME
  - <tex>E[M]=1</tex>\\ <tex>cov(N,M)=4</tex>
  - <tex>E[X]=\frac{1}{9}</tex>\\ <tex>var(X)=0.1133</tex>
  - 7,28


=====Guía 7: Procesos Poisson=====

  - 
    - 0,1429
    - 1 y 2
    - 1,7817
    - 0,218
    - Aumento en 1 las instalaciones
  - 
    - <tex>\sim Binom\left(p=\frac{2}{3},n=60\right)</tex>
    - 0,108
    - <tex>\sim Poisson(m=40)</tex>
  - 
    - <tex>e^{-6}</tex>
    - <tex>6e^{-6}</tex>
    - 1/3
  - 0,9901
  - 
    * Exponencial
      - <tex>e{}^{-0,2}</tex>
      - <tex>e{}^{-0,2}</tex>
      - No tiene memoria
    * Gamma
      - 0,9999
      - 0,6993 
  - 
    - <tex>f_{X,S}(x,s)=s\lambda^{3}e^{-\lambda(s+x)}</tex>
    - <tex>3\left(1-e^{-\lambda}\right)</tex>
  - <tex>E[T]=7,692</tex> minutos
  - 
    - 3/4
    - <tex>P(X=x)=\frac{3}{4^{x+1}}</tex> si <tex>x\text{\ensuremath{\in}N}</tex>
  - 
    - 0,0107
    - 0,0286
    - <tex>3\times10^{-7}</tex>
    - 1/2
    - 5/4
  - 
    - 0,1025
    - 0,2873
  - <tex>e{}^{-60}</tex>
  - 
    - 1/9
    - 5/9
  - 0,07985
  - <tex>E[T]=2</tex>, <tex>var(T)=4</tex>
  - 
    - 16,4
    - 0,0656
  - 1/4
  - 240
  - 
    - 0
    - 0,0148
    - 552 Kg
    - 331.200 Kg
  - FIXME
  - FIXME


=====Guía 8: Distribución normal, TCL=====

  - Trivial.
  - Trivial.
  - 0,676
  - 0,0917
  - 243,53
  - 
    - 491.95
    - IC: 394,26; 522,056
    - 0.6157
  - FIXME
  - No son independientes. <tex>cov(W,Z)\neq0</tex>
  - 
    - 0.9375
    - 0.9327
  - 
    - 0.245
    - 0.591
  - 0,5039
  - 0,4348
  - 
    - 0,0764
    - 0,318
    - 18.124,89
  - FIXME
  - 
    - 0,2209
    - 0,612
    - 0,2255; 0,6128
  - FIXME
  - 0,9545
  - FIXME
  - 0.023
  - 0
  - FIXME
  - 103
  - 1684
  - 64


=====Guía 9: Estadística Bayesiana=====

  - <tex>\pi(t|\mathbf{x})=6.76202\times0.0468644^{t}\times t^{5}\mbox{ si }t=\{1,\ldots,6\}</tex>\\ Media: 1,9356\\ Moda: 2
  - 
    - <tex>\pi(t|\mathbf{x})=\begin{cases}\frac{125}{62}\cdot\frac{1}{2\sqrt{2\pi}}\cdot e^{-0.5\left(\frac{12.1-\mu}{2}\right)^{2}} & \mbox{ si }\mu=10\\\frac{375}{62}\cdot\frac{1}{2\sqrt{2\pi}}\cdot e^{-0.5\left(\frac{12.1-\mu}{2}\right)^{2}} & \mbox{ si }\mu=14\end{cases}</tex>
    - <tex>P(Y>13)=0.2525</tex>
  - 
    - <tex>\pi(k|\mathbf{x})=\frac{6k-k^{2}}{35}</tex> si <tex>k=\{0,\ldots,6\}</tex>
    - 3
  - <tex>\pi(\theta|5)=\frac{1}{2}(5-\theta)\mbox{ si }\theta\in[3,5]</tex>
  - 
    - <tex>ab^{3}=3</tex>
    - <tex>\pi(t|\bar{\mathbf{x}})=26244t^{3}\mbox{ si }0<t<\frac{1}{9}</tex>\\ Media: <tex>\frac{4}{45}</tex>\\ Moda: <tex>\frac{1}{9}</tex>
  - <tex>\frac{13}{6}</tex>
  - <tex>\frac{7}{22}</tex>
  - FIXME
  - 
    - Si hay 0 personas irritadas de las 10 encuestadas, la primera opinión ("poca gente irritada") hará una buena estimación de <tex>p</tex>: <tex>\hat{p}=\frac{1}{21}</tex>.\\ Si hay 0 personas irritadas de las 10 encuestadas, la segunda opinión ("mucha gente irritada") hará una mala estimación de <tex>p</tex>: <tex>\hat{p}=\frac{10}{21}</tex>.\\ Si hay 10 personas irritadas de las 10 encuestadas, la primera opinión ("poca gente irritada") hará una mala estimación de <tex>p</tex>: <tex>\hat{p}=\frac{10}{21}</tex>.\\ Si hay 10 personas irritadas de las 10 encuestadas, la segunda opinión ("mucha gente irritada") hará una buena estimación de <tex>p</tex>: <tex>\hat{p}=\frac{20}{21}</tex>.
    - <tex>n\geq8989</tex>
  - 
    - Media: <tex>\frac{\nu}{\lambda}</tex>\\ Moda: <tex>\frac{\nu-1}{\lambda}</tex>
    - <tex>\pi(t|\bar{\mathbf{x}})\sim\Gamma\left(\lambda_{\Gamma}=n+\lambda,k_{\Gamma}=\left(\sum_{i=1}^{n}x_{i}\right)+\nu\right)</tex>\\ Media: <tex>\frac{1}{1+\frac{\lambda}{n}}\bar{x}+\frac{1}{1+\frac{\lambda}{n}}\frac{\nu}{n}</tex>
    - Moda: <tex>\frac{n+\nu-1}{\lambda+\sum_{i=1}^{n}x_{i}}</tex>
  - 
    - <tex>P\left(Y\geq2\right)\approx0.555</tex>
    - FIXME
  - 
    - <tex>\pi(t|\bar{\mathbf{x}})=295245t^{-6}\mbox{ si }t\geq9</tex>\\ Media: <tex>11.25</tex>\\ Moda: 9
    - <tex>\pi(t|\bar{\mathbf{x}})=38880t^{-6}\mbox{ si }t\geq6</tex>\\ Media: <tex>7.5</tex>\\ Moda: 6
  - 
    - <tex>\pi(t|\mathbf{x})=\frac{e^{-\frac{2}{25}\left(t-173\right)^{2}}}{5\sqrt{\frac{\pi}{2}}}\text{ si }\mu>0</tex>
    - Media: 173
    - FIXME
    - FIXME
    - FIXME


=====Guía 10: Estimación Puntual=====

  - Dado que <tex>ECM\left[\hat{\theta}_{1}\right]<ECM\left[\hat{\theta}_{2}\right]<ECM\left[\hat{\theta}_{3}\right]</tex>, el mejor estimador es <tex>\hat{\theta}_{1}</tex>.
  - 
    - <tex>\hat{\theta}_{1}</tex> es insesgado.\\ <tex>\hat{\theta}_{2}</tex> es sesgado.
    - FIXME
  - <tex>ECM\left[\hat{p}_{2}\right]<ECM\left[\hat{p}_{1}\right]</tex>, el estimador <tex>\hat{p}_{2}</tex> es preferible.
  - 
    - <tex>f_{\hat{\Theta}}(t)=\frac{n}{\Theta^{n}}\cdot t^{n-1}</tex>
    - <tex>B\left[\hat{\Theta}\right]=\Theta\left(\frac{n}{n+1}-1\right)</tex>
    - FIXME
    - FIXME
  - 
    - <tex>P\left(\bar{X}<0.25\right)=0.89435</tex>
    - <tex>P\left(S^{2}>0.577\right)=0.95</tex>
    - <tex>P\left(0.576\leq\hat{p}\leq0.764\right)=0.68244</tex>
    - FIXME
  - 
    - <tex>\hat{\mu}_{mv}=\bar{X}</tex>
    - <tex>\hat{p}_{mv}=\frac{\sum_{i=1}^{n}x_{i}}{n}</tex>
    - <tex>\hat{\theta}_{mv}=\max\left(x_{1},\ldots,x_{n}\right)</tex>
  - <tex>\hat{p}_ {mv}=\frac{2}{5}</tex>
  - <tex>\hat{M}_{mv}=50</tex>
  - 
    - -
    - FIXME
  - 
    - <tex>\hat{\mu}_{mv}=\frac{1}{n}\left[\sum_{i=1}^{n}x_{i}\right]</tex>\\ <tex>\sigma^{2}=\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}</tex>
    - FIXME
  - <tex>P(Y=1)=\frac{6}{11}</tex>
  - FIXME
  - 
    - <tex>\hat{\sigma^{2}} _{mv}=1.417</tex>
    - No existe <tex>\hat{\sigma^{2}} _{mv}</tex>
  - 
    * El emv de la media es <tex> 581.8</tex>
    * El emv de la varianza es <tex> 338491.24</tex>
    * El emv de la mediana es <tex> 403.2730</tex>
  - <tex> 0.64</tex>
  - FIXME
  - FIXME
  - <tex>\hat{\lambda} _{mv}=\frac{1}{15}</tex>
  - <tex>L \left(\lambda|\bar{x}\right)\propto\theta^{3}e^{-2\theta}</tex>\\ <tex>\hat{\theta} _{mv}=\frac{3}{2}</tex>
  - <tex> \hat{p}_{mv}=2\frac{r}{n}-\frac{1}{2}</tex>


=====Guía 11: Estimación por Intervalos=====

  - <tex> P\left(\frac{\mathbf{x}}{\sqrt{1-\alpha}}<\theta\right)=1-\alpha</tex>
  - 
    - <tex> Y=\frac{X_{(n)}}{\theta}</tex> es un pivote para <tex>\theta</tex>.
    - <tex> P\left(X_{(n)}<\theta<\frac{X_{(n)}}{\sqrt[n]{\alpha}}\right)=1-\alpha</tex>
    - <tex> k=\sqrt{10}</tex>
  - 
    - <tex> P\left(7.3845<\mu<8.6912\right)=0.95</tex>
    - <tex> n\geq38416</tex>
  - 
    - <tex> P\left(299836.6<\mu<299868.2\right)=0.95</tex>\\ El intervalo de confianza dado sólo dice lo siguiente: "si realizas otra medición de la velocidad de la luz, hay un 95\% de probabilidad de que <tex>\mu</tex> esté dentro de ese intervalo".
    - <tex>P\left(69.4855<\sigma<92.0106\right)=0.95</tex>
    - FIXME
  - 
    - <tex>X_{n+1}-\bar{X}_{n}\sim\mathcal{N}\left(0,\sigma^{2}\left[1-\frac{1}{n}\right]\right)</tex>
    - <tex>P\left(2.542<X_{6}<5.4758\right)=0.9</tex>
    - FIXME
  - 
    - <tex>P\left[0.07142<p<0.12878\right]\approx0.9</tex>
    - <tex>P\left[0.0778<p\right]\approx0.9</tex>
    - FIXME
  - 9604
  - 
    - <tex>P\left[2.71275<\lambda<7.8525\right]=0.9</tex>
    - <tex>P\left[2.78<5.4004\right]=0.9</tex>
  - <tex>P\left[0.00021667<\lambda\right]=0.95</tex>
  - 
    - <tex>P\left[0.15555<\lambda<1.02861\right]=0.95</tex>
    - <tex>P\left[0.15555<\lambda<1.02861\right]=0.95</tex>
  - 
    - <tex>P\left[0.4321<\Delta<56.31\right]=0.99</tex>
    - <tex>P\left[6.84<\Delta<49.9\right]=0.99</tex>
    - <tex>P\left[26.613<\Delta<30.128\right]=0.99</tex>
  - Proveedor 1.


=====Guía 12: Test de Hipótesis y Test de Bondad de Ajuste=====
  - <tex>P\left(\mbox{rechazar }H_{0}|\theta\right)=\frac{1}{35}\text{ si }\theta=3</tex>\\ <tex>P\left(\mbox{aceptar }H_{0}|\theta\right)=\frac{31}{35}\mbox{ si }\theta=4</tex>
  - 
    - <tex>P\left(\text{error tipo I}\right)=\frac{1}{3}</tex>\\ <tex>P\left(\text{error tipo II}\right)=\frac{1}{2}</tex>
    - <tex>P\left(\text{error tipo I}\right)=\frac{1}{3}</tex>\\ <tex>P\left(\text{error tipo II}\right)=\frac{1}{2}</tex>
    - FIXME
  - 
    - <tex>pot(\theta)=\begin{cases}\left(\frac{2.9}{\theta}\right)^{n} & \text{ si }\theta\in[3,4]\\1 & \text{ si }\theta\in(0;2.9)\\\left(\frac{2.9}{\theta}\right)^{n} & \mbox{ si }\theta\in(2.9,3)\\\left(\frac{2.9}{\theta}\right)^{n}+1-\left(\frac{4}{\theta}\right)^{n} & \text{ si }\theta\in(4,\infty)\end{cases}</tex>
    - <tex>NS=\left(\frac{2.9}{3}\right)^{n}</tex>
    - <tex>n\geq68</tex>
  - <tex>\delta(\mathbf{x})=\begin{cases}0 & \text{ si }2.3025\geq\mathbf{x}\\1 & \text{ si }2.3025<\mathbf{x}\end{cases}</tex>\\ <tex>P\left[\delta(\mathbf{x})=0|\mu=1.1\right]=0.876</tex>
  - <tex>\delta(\mathbf{x})=\begin{cases}1 & \mbox{ si }5<2.655583\\0 & \text{ si }5\geq2.655558\end{cases}</tex>\\ No rechazamos <tex>H_{0}</tex>.
  - 
    - F
    - V
    - F
    - F
    - V
    - F
    - F
    - V
  - 
    - <tex>\delta(\mathbf{x})=\begin{cases} 1 & \text{ si }\mu_{0}<\bar{X}-1.65\frac{\sigma}{\sqrt{n}}\\
0 & \text{ si }\mu_{0}\geq\bar{X}-1.65\frac{\sigma}{\sqrt{n}} \end{cases}</tex>
    - <tex>\beta(\mu)=1-\phi_{\bar{X}_{(\mu,\sigma)}}\left[\mu_{0}+1.65\frac{\sigma}{\sqrt{n}}\right]</tex>
  - FIXME
  - FIXME
  - FIXME
  - <tex>\delta(\mathbf{X})=\begin{cases} 1 & \text{ si }4\notin\left[\max\left\{ X_{i}\right\} ,1.3493\max\left\{ X_{i}\right\} \right]\\ 0 & \text{ si }4\in\left[\max\left\{ X_{i}\right\} ,1.3493\max\left\{ X_{i}\right\} \right]
\end{cases}</tex>
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - FIXME
  - No se puede rechazar <tex>H_{0}</tex>.\\ <tex>D^{2}=4.74</tex>
  - No se puede rechazar <tex>H_{0}</tex>.\\ <tex>D^{2}=3.26</tex>
  - No se puede rechazar <tex>H_{0}</tex>.\\ <tex>D^{2}=0.9</tex>
  - 
    - No se puede rechazar <tex>H_{0}</tex>.\\ <tex>D^{2}=10.408</tex>
    - No se puede rechazar <tex>H_{0}</tex>.\\ <tex>D^{2}=6.4</tex>
  - FIXME