Distribuciones de probabilidad multivariantes. 5,6 teoremas especiales.

5.77)

a)

Vamos con él. Hay que multiplicar la función de la que queremos hallar la esperanza por la función de densidad e integrar.

Como

0 <= y1 <= y2 <= 1

los límites de integración serán [0, 1] para Y1 y [y1, 1] para Y2

$$\begin{align}&E(Y_1)=\int_0^1\int_{y_1}^16y_1(1-y_2)dy_2dy_1=\\ &\\ &\\ &\int_0^1\left[ 6y_1\left(y_2-\frac{y_2^2}{2}\right)\right]_{y_1}^1dy_1 =\\ &\\ &\\ &\int_0^16y_1\left(1-\frac 12-y_1+\frac{y_1^2}{2}\right)dy_1 =\\ &\\ &\\ &\int_0^1\left(3y_1-6y_1^2+3y_1^3\right)dy_1 =\\ &\\ &\left[ \frac {3y_1^2}2 -2y^3+ \frac{3y_1^4}{4}\right]_0^1 =\frac 32-2+\frac 34=\frac 14\\ &\\ &\\ &\\ &\text{------------------------------------}\\ &\\ &\\ &\\ &E(Y^2)=\int_0^1\int_{y_1}^1 6y_2(1-y_2)dy_2dy_1=\\ &\\ &\\ &\int_0^1\int_{y_1}^1 (6y_2-6y_2^2)dy_2dy_1=\\ &\\ &\int_0^1\left[3y_2^2-2y_2^3  \right]_{y_1}^1dy_1=\\ &\\ &\\ &\int_0^1\left(3-2 -3y_2^2+2y_2^3  \right)dy_1=\\ &\\ &\left[ y_1-y_1^3+\frac{y_1^4}{2} \right]_0^1=1-1+\frac 12 = \frac 12\\ &\end{align}$$

b)

V(Y1) = E[(Y1)^2] - [E(Y1)]^2 = E[(Y1)^2] - (1/4)²

$$\begin{align}&E[(Y1)^2]= \int_0^1 \int_{y_1}^1 6 y_1^2(1-y_2)dy_2dy_1=\\ &\\ &\int_0^1\left[ 6y_1^2\left(y_2-\frac{y_2^2}{2}\right) \right]_{y_1}^1dy_1=\\ &\\ &\\ &\int_0^16y_1^2\left(1-\frac 12-y_1+\frac{y_1^2}{2}\right)dy_1=\\ &\\ &\int_0^1 (3y_1^2-6y_1^3+3y_1^4)dy_1=\\ &\\ &\\ &\left[ y_1^3-\frac {6y_1^4}{4}+\frac{3y_1^5}{5}\right]_0^1= 1-\frac 32+\frac 35 =\frac{1}{10}\\ &\\ &\end{align}$$

V(Y1) = 1/10 -1/16 = (8-5) / 80 = 3/80

V(Y2) = E[(Y2)²] - (1/2)²

$$\begin{align}&E[(Y2)^2]= \int_0^1 \int_{y_1}^1 6 y_2^2(1-y_2)dy_2dy_1=\\ &\\ &\\ &\int_0^1 \left[ 2y_2^3-\frac{6y_2^4}{4} \right]_{y_1}^1dy_1=\\ &\\ &\\ &\int_0^1 (2-\frac 32-2y_1^3+\frac{6y_1^4}{4})dy_1=\\ &\\ &\left[\frac{y_1}{2}-\frac{2y_1^4}{4}+\frac{3y_1^5}{10} \right]_0^1=\frac 12-\frac 12+\frac {3}{10} =\frac{3}{10}\end{align}$$

V(Y2) = 3/10 -1/4 = (6-5)/20 = 1/20

c) E(Y1-3Y2)=

Aplicando un teorema que hay

= E(Y1) - 3E(Y2) =(1/4) - 3(1/2) = (1-6)/4 = -5/4

Y eso es todo.