Hello, Can someone tell me how to go about getting the following questions started i dont need to know the answer but at the same time I havent the slightest clue about what they are asking me to do. Please see attachment thanks much.
Hello, Can someone tell me how to go about getting the following questions started i dont need to know the answer but at the same time I havent the slightest clue about what they are asking me to do. Please see attachment thanks much.
Question#1
The ‘probability of false alarm’ is given by…
$\displaystyle p_{FA}= \int_{\eta}^{\infty} \frac{x^{\frac{p}{2}-1}\cdot e^{-\frac{x}{2}}}{\Gamma (\frac{p}{2})\cdot 2^{\frac{p}{2}}}\cdot dx$ (1)
… and setting $\displaystyle p=6$ is…
$\displaystyle p_{FA}= \int_{\eta}^{\infty} \frac{x^{2}\cdot e^{-\frac{x}{2}}}{2^{4}}\cdot dx= \frac{1}{2}\cdot \int_{\eta}^{\infty} t^{2}\cdot e^{-t}\cdot dt = e^{-\frac{\eta}{2}}(1+\frac{\eta}{2} + \frac{\eta^{2}}{8})$ (2)
The problem is to find $\displaystyle \eta$ for which is $\displaystyle p_{FA}=.001,p_{FA}=.01,p_{FA}=.1$. The method suggested in textbook is the bisection method, which is very inaccurate and requires long and tedious computation. Much better is the Newton-Raphson method, which solves an equation in the form $\displaystyle f(\eta)=0$ via the iterative procedure…
$\displaystyle \eta_{n+1}= \eta_{n}- \frac {f(\eta_{n})}{f^{'}(\eta_{n})}$ (3)
In our case is…
$\displaystyle f(\eta)= e^{-\frac{\eta}{2}}\cdot (1+\frac{\eta}{2} + \frac{\eta^{2}}{8}) - p_{FA}$
$\displaystyle f^{'}(\eta) = -\frac {1}{16}\cdot \eta^{2}\cdot e^{-\frac{\eta}{2}}$ (4)
In all cases the initial value is $\displaystyle \eta_{0} = 10$…
For $\displaystyle p_{FA}=.1$ in $\displaystyle 4$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 10.64464$…
For $\displaystyle p_{FA}=.01$ in $\displaystyle 6$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 16.81189$…
For $\displaystyle p_{FA}=.001$ in $\displaystyle 8$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 22.45774$…
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$