Find the derivatives with the respect to \(x\) of the following.
\(F(x|x\ge 0)= 1- e^{-\lambda x}\)
\(F(x|b> a)= \frac{x-a}{b-a}\)
\(F(x|a< x \le c\le b)=\frac{(x-a)^2}{(b-a)(c-a)}\)
\(F(x| a\le c<x<b)=1-\frac{(b-x)^2}{(b-a)(c-a)}\)
Solve the following definite and indefinite integrals
\(\int_0^{10}3x^3dx\)
\(\int_0^x x \lambda e^{-\lambda x}dx\)
\(\int_0^.5 \frac{1}{b-a}dx\)
\(\int_0^x x\frac{1}{\Gamma (\alpha) \beta ^\alpha}x^{\alpha -1}e^{-\beta x}dx\)
Hint: the last part of the equation is beginning with the gamma function is a Gamma probability distribution function. Try rearranging the terms to integrate another Gamma distribution out of the integral, as pdfs must integrate to 1.
With the following matrix, \[\mathbf{X} = \left[\begin{array}{rrr}1 & 2 & 3\\3 & 3 & 1\\4 & 6 & 8\end{array}\right]\]
Invert it using Gaussian row reduction.
Find the determinant.
Conduct LU decomposition
Multiply the matrix by it’s inverse.