- math
    - x<c(12,5,13)
    - cumsum(x)
        - 12 17 30
    - cumprod(x)    # mul
        - 12 60 780
    - z
        - 1 2
        - 5 3
        - 6 2
    - min(z[,1],z[,2])
        - 1
    - pmin(z[,1],z[,2])    # by row
        - 1 3 2
    - pmin(z[1,],z[2,],z[3,])    # by col
        - 1 2
    - nlm(function(x) return(x^2-sin(x)),8)
        - minimum    # -0.23
        - estimate    # 0.45    when x=0.45 functions's minimum is -0.23
        - iterations    # 5    using Newton-Raphson method
    - calculus
    - D(expression(exp(x^2)),"x")    # derivative
        - exp(x^2)*(2*x)
    - integrate(function(x) x^2,0,1)
        - 0.333333 with aboslute error < 3.7e-15

- distribution
    - dnorm        dchiq    dbinom
    - pnorm        pchisq    pbinom
    - qnorm        qchisq    qbinom
    - rnorm        rchisq    rbinom
    - mean(rchisq(1000,df=2))
        - 1,938179
    - qchisq(0.95,2)
        - 5.991465
    - qchisq(c(0.5,0.95),df=2)
        - 1.386294 5.991465

- rank
    - y
    - V1 V2
    - def 2
    - ab 5
    - zzzz 1
    - r<-order(y$V2)    # 3 1 2
    - z<-y[r,]
    - z
        - V1 V2
        - zzzz 1
        - def 2
        - ab 5
    - d
        - kids ages
        - Jack 12
        - Jill 10
        - Billy 13
    - d[order(d$kids),]
        - kids ages
        - Billy 13
        - Jack 12
        - Jill 10
    - d[order(d$ages),]
        - kids ages
        - Jill 10
        - Jack 12
        - Billy 13
    - x<-c(13,5,12,5)
    - rank(x)    # 4.0 1.5 3.0 1.5    5 occurs twice with rank 1 and 2, so mean is 1.5

- algebra
    - crossprod(1:3,c(5,12,13))    # 68=1*5+2*12+3*13, dot product
    - a<-matrix(c(1,1,-1,1),nrow=2,ncol=2)
    - b<-c(2,4)
    - solve(a,b)    # solve equation    x1+x2=2 -x1+x2=4
        - 3 1
    - solve(a)    # inverse matrix
        - 0.5 0.5
        - -0.5 0.5
    - m
        - 1 2
        - 7 8
    - dm<-diag(m)
    - dm
        - 1 8
    - diag(m)
        - 1 0
        - 0 8
    - diag(3)
        - 1 0 0
        - 0 1 0
        - 0 0 1
    - m
        - 1 2 3
        - 4 5 6
        - 7 8 9
    - sweep(m,1,c(1,4,7),"+")    # 1 means by row
        - 2 3 4
        - 8 9 10
        - 14 15 16

- set
    - x<-c(1,2,5)
    - y<-c(5,1,8,9)
    - union(x,y)    # 1 2 5 8 9
    - intersect(x,y)    # 1 5
    - setdiff(x,y)    # 2    in x but not in y
    - setdiff(y,x)    # 8 9 in y but not in x
    - setequal(x,y)    # FALSE
    - setequal(x,c(1,2,5))    # TRUE
    - 2%in%x    # TRUE
    - 2%in%y    #FALSE
    - choose(5,2)    # 10 subset number like combination(5,2)
    - symdiff<-function(a,b) {
        - a<-setdiff(x,y)
        - b<-setdiff(y,x)
        - return(union(a,b))
    - }
    - "%subsetof%"<-function(u,v) {    # is u a subset of v
        - return(setequal(intersect(u,v),u))
    - }
    - c(3,8)%subsetof%1:10    # TRUE
    - c(3,8)%subsetof%5:10    # FALSE
    - combn(1:3,2,sum)    # 3 4 5 all 2-length subset's sum {1,2} {1,3} {2,3}

- simulation
    - x<-rbinom(10000,5,0.5)    # 10000 random variables
    - mean(x>=4)    # 0.18829
    - sum<-0
    - nreps<-100000
    - for (i in 1:nreps) {
        - xy<-rnorm(2)    # 2 N(0,1)s
        - sum<-sum+max(xy)
    - }
    - print(sum/nreps)
    - emax<-function(nreps) {
        - x<-rnorm(2*nreps)
        - maxxy<-pmax(x[1:nreps],x[(nreps+1):(2*nreps)])
        - return(mean(maxxy))
    - }
    - set.seed(8888)