Week 1 Math Homework (1)

Derivatives

q1. F(x|x≥0)=1−e^(−λx)

#x[x>=0]
q1<- expression(1-exp^(lambda*x))
q1_a<-D(q1,'x')

print (q1_a)
## -(exp^(lambda * x) * (log(exp) * lambda))

Q2. F(x|b>a)=(x−a)/(b−a)

#x[b>a]
q2 = expression((x−a)/(b−a))
q2_a <- D(q2, "x")
print (q2_a) 
## 1/(b - a)

Q3: F(x|a<x≤c≤b)=(x−a)2/(b−a)(c−a)

#x[a<x<=c<=b]
q3 = expression((x-a)^2/(b-a)*(c-a))
q3_a <- D(q3,'x')
print (q3_a)
## 2 * (x - a)/(b - a) * (c - a)

Q4: F(x|a≤c<x<b)=1−(b−x)^2/(b−a)(c−a)

q4 = expression(1-(b-x)^2/(b-a)*(c-a))
q4_a <- D(q4,'x')
print (q4_a)
## 2 * (b - x)/(b - a) * (c - a)

Q5: ∫0/10 3x^3 dx

q5 <- function (x) {3*x^3}
integrate(q5, lower = 0, upper = 10)
## 7500 with absolute error < 8.3e-11

Q6: ∫o/x xλe−λx dx

Let u=x and dv=e^-x

du= 1dx and v= intg of e^-x du - (intg of vdu)

1 - (-e^-x +c) = 1 + e^-x +c

lambda <- 1
q6<- function (x) x*lambda
q6_a <- Deriv(q6)
print (q6_a)
## function (x) 
## lambda

Q7:∫0/.5 1/b−a dx

antiDeriv = x/(b-a)+c

q7<- antiD(1/(b-a)~x)
print (q7)
## function (x, C = 0, b, a) 
## 1/(b - a) * x + C

q8<- ∫0/x x(1/Γ(α)β^α * x^α−1*exp(−βx) dx

rewritten as: (1/RhoalphaBetaalpha)(x^alpha-1)e-Beta(x) AntiDeriv = x{alpha+1}/alphabetagamma{alpha}e^{beta}

q8 = antiD(x^{alpha+1}/alpha*beta*gamma^{alpha}*e^{beta}~x)
print (q8)
## function (x, alpha, beta, gamma, e, C = 0) 
## {
##     numerical_integration(.newf, .wrt, as.list(match.call())[-1], 
##         formals(), from, ciName = intC, .tol)
## }
## <environment: 0x000000001e092de0>

Linear Algebra

Create Matrix [123] x=[331] [468]

LA_matrix <- matrix (c(1,2,3,3,3,1,4,6,8), nrow =3, byrow = TRUE)

Prep work creating the Identity Matrix

LA_I <- cbind(LA_matrix, diag (3))
print (LA_I)
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    2    3    1    0    0
## [2,]    3    3    1    0    1    0
## [3,]    4    6    8    0    0    1

Q9: Invert it using Gaussian row reduction (Gaussian Elimination)

LA_I[2,] <- -3 * LA_I[1,] + LA_I[2,]    
LA_I[3,] <- -4 * LA_I[1,] + LA_I[3,]   
LA_I[2,] <- -(1/3) * LA_I[2,]         
LA_I[1,] <- -2 * LA_I[2,] + LA_I[1,]    
LA_I[3,] <- 2 * LA_I[2,] + LA_I[3,]     
LA_I[3,] <- (3/4) * LA_I[3,]      
LA_I[1,] <- (7/3) * LA_I[3,] + LA_I[1,] 
LA_I[2,] <- -(8/3) * LA_I[3,] + LA_I[2,]

inverseLA <- LA_I[,-(1:3)]

print(inverseLA)
##      [,1] [,2]  [,3]
## [1,] -4.5 -0.5  1.75
## [2,]  5.0  1.0 -2.00
## [3,] -1.5 -0.5  0.75

Q10: Find the determinant.

q10 <- det(LA_matrix)
print (q10)
## [1] -4

Q11. Conduct LU decomposition

LU_matrix <- lu(LA_matrix)
expandLU_matrix <- expand (LU_matrix)
expandLU_matrix$L
## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
##      [,1]       [,2]       [,3]      
## [1,]  1.0000000          .          .
## [2,]  0.7500000  1.0000000          .
## [3,]  0.2500000 -0.3333333  1.0000000
expandLU_matrix$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

Q12. Multiply the matrix by it’s inverse

inverseLA %*% LA_matrix
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1