Homework 1
Derivatives
Find the derivatives with the respect to x of the following.
- F(x|x≥0)=1−e−λx
f1 <- function(x){ 1-exp(-lx) }
Deriv(f1)## function (x)
## 0
- F(x│b>a)=(x-a)/(b-a)
f2 <- function(x){ (x-a)/(b-a) }
Deriv(f2)## function (x)
## 1/(b - a)
- F(x│a<x≤c≤b)=(x-a)^2/(b-a)(c-a)
f3 <- function(x){ (x-a)^2 / (b-a)(c-a) }
Deriv(f3)## function (x)
## 2 * ((x - a)/(b - a)(c - a))
- F(x│a≤c<x<b)=1-(b-x)^2/((b-a)(c-a))
f4 <- function(x){ 1-((b-x)^2 / (b-a)(c-a)) }
Deriv(f4)## function (x)
## 2 * ((b - x)/(b - a)(c - a))
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Integrals
Solve the following definite and indefinite integrals.
- ∫_0^10〖 3x^3 dx〗
f5 <- antiD(3*x^3~x, x=10)
f5## function (x = 10, C = 0)
## 3/4 * x^4 + C
- ∫_(0)(x)〖xλe(-λx) dx〗
f6 <- function(x) { x*antiD(exp(-x)~x)-(1-exp(-x)) }
antiD(f6~x)## function (x, C = 0, f6)
## f6 * x + C
- ∫_(0)^(.5)〖 1/(b-a) dx〗
f7 <- antiD((1/(b - a)) ~x, x = 0.5)
f7## function (x = 0.5, C = 0, b, a)
## (1/(b - a)) * x + C
- ∫_(0)^(x) x 1/Γ(α)βα x^(α-1) e^(-βx ) dx
f8 <- antiD(x*(1/(Γ(α)*B^α))*x^(α−1)*e^(−Bx)~x)
f8## function (x, a, B, e, Bx, C = 0)
## {
## numerical_integration(.newf, .wrt, as.list(match.call())[-1],
## formals(), from, ciName = intC, .tol)
## }
## <environment: 0x000000002207c2e0>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 out10 of the integral, as pdfs must integrate to 1.
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Linear Algebra
With the following matrix,
\[ x = [(1,2,3)(3,3,1)(4,6,8)] \]
- Invert it using Gaussian row reduction.
m1 <- matrix(c(1, 3, 4, 2, 3, 6, 3, 1, 8), ncol = 3)
solve(m1)## [,1] [,2] [,3]
## [1,] -4.5 -0.5 1.75
## [2,] 5.0 1.0 -2.00
## [3,] -1.5 -0.5 0.75
- Find the determinant.
m2 <- matrix(c(1, 3, 4, 2, 3, 6, 3, 1, 8), ncol = 3)
determinant(m2)## $modulus
## [1] 1.386294
## attr(,"logarithm")
## [1] TRUE
##
## $sign
## [1] -1
##
## attr(,"class")
## [1] "det"
- Conduct LU decomposition.
m3 <- matrix(c(1, 3, 4, 2, 3, 6, 3, 1, 8), ncol = 3)
lu(m3)## 'MatrixFactorization' of Formal class 'denseLU' [package "Matrix"] with 4 slots
## ..@ x : num [1:9] 4 0.75 0.25 6 -1.5 ...
## ..@ perm : int [1:3] 3 2 3
## ..@ Dimnames:List of 2
## .. ..$ : NULL
## .. ..$ : NULL
## ..@ Dim : int [1:2] 3 3 expand(lu(m3))## $L
## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
## [,1] [,2] [,3]
## [1,] 1.0000000 . .
## [2,] 0.7500000 1.0000000 .
## [3,] 0.2500000 -0.3333333 1.0000000
##
## $U
## 3 x 3 Matrix of class "dtrMatrix"
## [,1] [,2] [,3]
## [1,] 4.0000000 6.0000000 8.0000000
## [2,] . -1.5000000 -5.0000000
## [3,] . . -0.6666667
##
## $P
## 3 x 3 sparse Matrix of class "pMatrix"
##
## [1,] . . |
## [2,] . | .
## [3,] | . .
- Multiply the matrix by it’s inverse.
m4 <- matrix(c(1, 3, 4, 2, 3, 6, 3, 1, 8), ncol = 3)
m4%*%solve(m4)## [,1] [,2] [,3]
## [1,] 1.000000e+00 -2.220446e-16 4.440892e-16
## [2,] -4.440892e-16 1.000000e+00 2.220446e-16
## [3,] 0.000000e+00 0.000000e+00 1.000000e+00Please email to: kleber.perez@live.com for any suggestion.
Other links
Useful links:
-- Sample SQL
SELECT FORMAT(AVG(alt),2) as 'Avg. altitude'
FROM airports
WHERE faa IN ('EWR','LGA','JFK');