Exercises

Minimal MSE estimate from joint PDF

Here is the joint PDF of $X$ and $Y$:

$$f_{X,Y}=\{\begin{array}{cc}8xy& 0\le y\le x\le 1\\ 0& \text{otherwise}\end{array}$$

Find the minimal MSE estimate of $X$ in terms of $Y$.

What is the estimate of $X$ when $Y=0.2$? When $Y=0.8$?

Answer

${\displaystyle {\hat{x}}_M(y)=\frac{2}{3}\cdot \frac{1-y^3}{1-y^2}}$

${\hat{x}}_M(0.2)=0.6889{\hat{x}}_M(0.8)=0.9037$

Line of minimal MSE given joint PDF

Here is the joint PDF of $X$ and $Y$:

$$f_{X,Y}=\{\begin{array}{cc}8xy& 0\le y\le x\le 1\\ 0& \text{otherwise}\end{array}$$

Find the line giving the linear MSE estimate of $X$ in terms of $Y$.

What is the expected error of this line, $e_{L_{\min }}$?

What is the estimate of $X$ when $Y=0.2$? When $Y=0.8$?

Answer

${\hat{X}}_{L_{\min }}(Y)=0.3637Y+0.6060$

$e_{L_{\min }}=0.02020$

${\hat{x}}_{L_{\min }}(0.2)=0.67874{\hat{x}}_{L_{\min }}(0.8)=0.89696$