Mathematics is important for coding

What mathematics is needed for coding in R and Python?

Mathematics plays a crucial role in coding and computer programming, and its importance cannot be overstated. Here are several reasons why mathematics is important for coding:

Algorithms and Problem-Solving: Coding involves the development of algorithms, which are step-by-step instructions for solving a problem or performing a task. Mathematics provides the foundation for creating efficient algorithms and solving complex problems.

Logic and Boolean Algebra: Programming relies heavily on logical reasoning and Boolean algebra. Mathematical concepts such as logic gates, truth tables, and Boolean operations are fundamental to writing code that makes decisions and processes data logically.

Data Structures: Data structures, such as arrays, linked lists, trees, and graphs, are essential components of computer programs. Understanding mathematical concepts, like set theory and graph theory, helps in designing and manipulating these data structures effectively.

Complexity Analysis: Mathematics is instrumental in analyzing the time and space complexity of algorithms. This analysis helps programmers evaluate the efficiency of their code and make improvements to optimize performance.

Numerical Operations: Many programming tasks involve numerical operations, including arithmetic, calculus, and linear algebra. These mathematical concepts are vital for tasks such as scientific computing, data analysis, and simulations.

Cryptography and Security: In the realm of cybersecurity and cryptography, advanced mathematical concepts like number theory are crucial for creating secure encryption algorithms and protecting data.

Machine Learning and Artificial Intelligence: Mathematics, particularly statistics and linear algebra, is foundational to machine learning and artificial intelligence. These fields heavily rely on mathematical models, equations, and algorithms to make predictions, classify data, and optimize decision-making processes.

Graphics and Game Development: In game development and computer graphics, mathematics is essential for rendering 3D objects, simulating physics, handling transformations, and creating realistic visual effects.

Signal Processing: In fields like audio processing and image processing, mathematical techniques, including Fourier transforms and convolution, are used to manipulate and analyze signals and data.

Databases and SQL: When working with databases and querying data using SQL (Structured Query Language), mathematical set operations and relational algebra concepts come into play.

Coding Challenges and Competitions: Many coding competitions, such as those on platforms like LeetCode and HackerRank, involve solving mathematical and algorithmic problems. Strong mathematical skills are an asset in such competitions.

Debugging and Troubleshooting: The ability to think analytically and systematically, which is often nurtured through mathematical training, can be highly beneficial when debugging and troubleshooting code.

In summary, mathematics provides the intellectual tools and problem-solving techniques necessary for effective coding and computer programming. While not every coding task requires advanced mathematics, a solid foundation in mathematical concepts and logical reasoning can greatly enhance a programmer's ability to write efficient, reliable, and innovative code. Moreover, as technology continues to advance, the intersection of mathematics and coding will become increasingly important in various domains, from artificial intelligence to cybersecurity.

A number of people do not know that Python helps to learn algebra, calculus, and matrix analysis. 

R Programming Code:

> x<-seq(0,10,0.2)

> window(7,4)

> par(mfrow=c(1,2)) 


How to write exponential and log in R and Python? 

> y <- exp(x)
> plot (x~y,type='l',main="Exponential")
> y <- log(x)
> plot(y~x,type="l",main="Logarithmic")
> plot (y~x,type='l',main="Exponential")
> y <- exp(x)
> plot (y~x,type='l',main="Exponential")
> #Trigonometric functions
> windows(7,7)

> par(mfrow=c(2,2))
> x <- seq(0,2*pi,2*pi/100)
> y1 <- cos(x)
> y2 <- sin(x)
> y3 <- tan(x)
> plot(y1~x,type="l",main="cosine")

> plot(y1~x,type="l",main="cosine")
> plot(y2~x,type="l",main="sine")
> plot(y1~x,type="l",main="tangent")
> plot(y3~x,type="l",ylim=c(-3,3),main="tangent")
> plot(y3~x,type="l",main="tangent")
> #power laws
> x<-seq(0,10,.01)
> y
 [1]     1.000000     1.221403     1.491825
 [4]     1.822119     2.225541     2.718282
 [7]     3.320117     4.055200     4.953032
[10]     6.049647     7.389056     9.025013
[13]    11.023176    13.463738    16.444647
[16]    20.085537    24.532530    29.964100
[19]    36.598234    44.701184    54.598150
[22]    66.686331    81.450869    99.484316
[25]   121.510418   148.413159   181.272242
[28]   221.406416   270.426407   330.299560
[31]   403.428793   492.749041   601.845038
[34]   735.095189   897.847292  1096.633158
[37]  1339.430764  1635.984430  1998.195895
[40]  2440.601978  2980.957987  3640.950307
[43]  4447.066748  5431.659591  6634.244006
[46]  8103.083928  9897.129059 12088.380730
[49] 14764.781566 18033.744928 22026.465795
> y=x^0.5

How to plot sequence in python?
> plot(x,y,type="l",main="0<b<1")
> y <- x
> plot(x,y,type="l",main="b=1")
> y <- xˆ2

> y <- x^2
> plot(x,y,type="l",main="b>1")
> y <- 1/x
> plot(x,y,type="l",main="b<0")
> #polynomial function 
> x<-seq(0,10,.01)
> y1 <- 2+5*x-0.2*x^2
> y2 <- 2+5*x-0.4*x^2
> y3 <- 2+4*x-0.6*x^2+0.04*x^3
> y4<- 2+4*x+2*x^2-0.6*x^3+0.04*x^4
> par(mfrow=c(2,2))
> plot(x,y1,type="l",ylab="y",main="decelerating")
> plot(x,y2,type="l",ylab="y",main="humped")
> plot(x,y3,type="l",ylab="y",main="inflection")
> plot(x,y4,type="l",ylab="y",main="local maximum")
> #power set 
> x <- seq(0,1,0.01)
> y <- xˆ0.5
How to write squareroot x
> y <- x^0.5
> plot(x,y,type="l",main="0<b<1")
> y <- x
> plot(x,y,type="l",main="b=1")
> y <- x^2
> plot(x,y,type="l",main="b>1")
> y <- 1/x
> plot(x,y,type="l",main="b<0")
> #Polynomial functions
> par(mfrow=c(2,2))
> y1 <- x/(2+5*x)
> y2 <- 1/(x-2+4/x)
> y3 <- 1/(xˆ2-2+4/x)

> y3 <- 1/(x^2-2+4/x)
> plot(x,y1,type="l",ylab="y",main="Michaelis-Menten")
> plot(x,y2,type="l",ylab="y",main="shallow hump")
> plot(x,y3,type="l",ylab="y",main="steep hump")
> #Gamma function
> par(mfrow=c(1,1))
> t <- seq(0.2,4,0.01)
> plot(t,gamma(t),type="l")
> abline(h=1,lty=2)
> abline(h=4.8,lty=2)
> #Asymptotic functions
> (1/44.44 - 1/70.59)/(1/0.2 - 1/0.6)
[1] 0.002500781
> #Sigmoid (S-shaped) functions
> par(mfrow=c(2,2))
> x <- seq(0,10,0.1)
> y <- 100/(1+90*exp(-1*x))
> plot(x,y,type="l",main="three-parameter logistic")
> #four-parameter logistic function
> y <- 20+100/(1+exp(0.8*(3-x)))
> plot(x,y,ylim=c(0,140),type="l",main="four-parameter logistic")
> #Gompertz growth model,
> x <- -200:100
> y <- 100*exp(-exp(0.02*x))
> plot(x,y,type="l",main="negative Gompertz")
> x <- 0:100
> y <- 50*exp(-5*exp(-0.08*x))
> plot(x,y,type="l",main="positive Gompertz")
> #Biexponential model
> a <- 10
> b <- -0.8
> c <- 10
> d <- -0.05
> y <- a*exp(b*x)+c*exp(d*x)
> plot(x,y,main="+ - + -",type="l")
> a <- 10
> b <- -0.8
> c <- 10
> d <- 0.05
> y <- a*exp(b*x)+c*exp(d*x)
> plot(x,y,main="+ - + +",type="l")
> a <- 200
> b <- 0.2
> c <- -1
> d <- 0.7
> y <- a*exp(b*x)+c*exp(d*x)
> 
> plot(x,y,main="+ + - +",type="l")
> a <- 200
> b <- 0.05
> c <- 300
> d <- -0.5
> y <- a*exp(b*x)+c*exp(d*x)
> plot(x,y,main="+ + + -",type="l")
> #Probability functions
> par(mfrow=c(1,1))
> x <- 0:6
> plot(x,factorial(x),type="s",main="factorial x",log="y")
> plot(x,factorial(x)/(x-2),type="s",main="factorial x",log="y")

> choose(8,3)
[1] 56
> plot(0:8,choose(8,0:8),type="s",main="binomial coefficients")
> curve(pnorm(x),-3,3)
> arrows(-1,0,-1,pnorm(-1),col="red")
> arrows(-1,pnorm(-1),-3,pnorm(-1),col="green")
> pnorm(-1)
[1] 0.1586553
> #probability density
> curve(dnorm(x),-3,3)
> #Normal distribution
> par(mfrow=c(2,2))
> x <- seq(-3,3,0.01)
> y <- exp(-abs(x))
> plot(x,y,type="l",main= "x")
> y <- exp(-abs(x)ˆ2)

> y <- exp(-abs(x)^2)
> plot(x,y,type="l",main= "x^2")
> y <- exp(-abs(x)^3)
> plot(x,y,type="l",main= "x^3")
> y <- exp(-abs(x)^8)
> plot(x,y,type="l",main= "x^8")
> pnorm(-1.25)
[1] 0.1056498
> pnorm(1.875)
[1] 0.9696036
> 1-pnorm(1.875)
[1] 0.03039636
> pnorm(1.25)-pnorm(-0.625)
[1] 0.6283647
> par(mfrow=c(2,2))
> lambda <- 2
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 2.0")
> lambda <- 3.3
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 3.3")
> lambda <- 3.5
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 3.5")
> lambda <- 4
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 4")
> lambda <- 4
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 4.0")
> lambda <- 3.5
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 3.5")
> lambda <- 3
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 3")
> lambda <- 2
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 2")
> lambda <- 12
> x <- numeric(40)
> x[1] <- 0.6
> for (t in 2 : 40) x[t] <- lambda * x[t-1] * (1 - x[t-1])
> plot(1:40,x,type="l",ylim=c(0,1),ylab="population",
+      xlab="time",main="lambda = 12")
> numbers <- function (lambda) {
+     x <- numeric(400)
+     x[1] <- 0.6
+     for (t in 2 : 400) x[t] <- lambda * x[t-1] * (1 - x[t-1])
+     x[381:400] }
> par(mfrow=c(1,1))
> plot(c(2,4),c(0,1),type="n",xlab="lambda",ylab="population")
> for(lam in seq(2,4,0.01))
+     points(rep(lam,20),sapply(lam,numbers),pch=16,cex=0.5,col="blue")
> plot(0:100,0:100,type="n",xlab="",ylab="")
> x <- y <- 50
> points(50,50,pch=16,col="red",cex=1.5)
> for (i in 1:10000) {
+     xi <- sample(c(1,0,-1),1)
+     yi <- sample(c(1,0,-1),1)
+     lines(c(x,x+xi),c(y,y+yi),col="blue")
+     x <- x+xi
+     y <- y+yi
+     if (x>100 | x<0 | y>100 | y<0) break
+ }
> y <- c(8,3,5,7,6,6,8,9,2,3,9,4,10,4,11)
> mean(y)
[1] 6.333333
> median(y)
[1] 6
> mode(y)
[1] "numeric"
> range(y)
[1]  2 11
> var(y)
[1] 7.809524
> quantile(y)
  0%  25%  50%  75% 100% 
 2.0  4.0  6.0  8.5 11.0 
> cumsum(y)
 [1]  8 11 16 23 29 35 43 52 54 57 66 70 80 84 95
> colMeans(y)
How to write matrix code?
> counts <- rnbinom(10000,mu=0.92,size=1.1)
> counts[1:30]
 [1] 1 2 0 1 0 4 1 0 0 0 4 1 1 1 1 0 0 1 0 0 0 3 0
[24] 0 3 1 0 0 0 0
> X <- matrix(c(1,0,0,0,1,0,0,0,1),nrow=3)
> X
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1
> class(X)
[1] "matrix" "array" 
> attributes(X)
$dim
[1] 3 3

> vector <- c(1,2,3,4,4,3,2,1)
> V <- matrix(vector,byrow=T,nrow=2)
> V
     [,1] [,2] [,3] [,4]
[1,]    1    2    3    4
[2,]    4    3    2    1
> dim(vector) <- c(4,2)
> dim(vector)
[1] 4 2
> is.matrix(vector)
[1] TRUE
> vector
     [,1] [,2]
[1,]    1    4
[2,]    2    3
[3,]    3    2
[4,]    4    1
> (vector <- t(vector))
     [,1] [,2] [,3] [,4]
[1,]    1    2    3    4
[2,]    4    3    2    1
> X <- matrix(rpois(20,1.5),nrow=4)
> X
     [,1] [,2] [,3] [,4] [,5]
[1,]    2    0    1    2    0
[2,]    1    1    2    1    4
[3,]    0    2    1    2    2
[4,]    2    1    0    2    2
> X <- rbind(X,apply(X,2,mean))
> X
     [,1] [,2] [,3] [,4] [,5]
[1,] 2.00    0    1 2.00    0
[2,] 1.00    1    2 1.00    4
[3,] 0.00    2    1 2.00    2
[4,] 2.00    1    0 2.00    2
[5,] 1.25    1    1 1.75    2
> X <- cbind(X,apply(X,1,var))
> X
     [,1] [,2] [,3] [,4] [,5]    [,6]
[1,] 2.00    0    1 2.00    0 1.00000
[2,] 1.00    1    2 1.00    4 1.70000
[3,] 0.00    2    1 2.00    2 0.80000
[4,] 2.00    1    0 2.00    2 0.80000
[5,] 1.25    1    1 1.75    2 0.20625
> colnames(X) <- c(1:5,"variance")
> X
        1 2 3    4 5 variance
[1,] 2.00 0 1 2.00 0  1.00000
[2,] 1.00 1 2 1.00 4  1.70000
[3,] 0.00 2 1 2.00 2  0.80000
[4,] 2.00 1 0 2.00 2  0.80000
[5,] 1.25 1 1 1.75 2  0.20625
> rownames(X) <- c(1:4,"mean")
> X
        1 2 3    4 5 variance
1    2.00 0 1 2.00 0  1.00000
2    1.00 1 2 1.00 4  1.70000
3    0.00 2 1 2.00 2  0.80000
4    2.00 1 0 2.00 2  0.80000
mean 1.25 1 1 1.75 2  0.20625
> rowmatrix <- mat[2, , drop = FALSE]
How to use appply function?
> apply(X,1,function(x) x^2+x)
         1     2    3    4      mean
1        6  2.00 0.00 6.00 2.8125000
2        0  2.00 6.00 2.00 2.0000000
3        2  6.00 2.00 0.00 2.0000000
4        6  2.00 6.00 6.00 4.8125000
5        0 20.00 6.00 6.00 6.0000000
variance 2  4.59 1.44 1.44 0.2487891
> sapply(3:7, seq)
[[1]]
[1] 1 2 3

[[2]]
[1] 1 2 3 4

[[3]]
[1] 1 2 3 4 5

[[4]]
[1] 1 2 3 4 5 6

[[5]]
[1] 1 2 3 4 5 6 7

> a <- seq(0.01,0.2,.005)
> plot(a,sapply(a,sumsq),type="l")

> # creat the vector
> x<-1:30
> x
 [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
[16] 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
> y<-29:1
> y
 [1] 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
[16] 14 13 12 11 10  9  8  7  6  5  4  3  2  1
> z<-(1:30,29:1)

> z<-c(1:30,29:1)
> z
 [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
[16] 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
[31] 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
[46] 14 13 12 11 10  9  8  7  6  5  4  3  2  1
> tep<-c(4,6,3)
> tep
[1] 4 6 3
> 3^tep
[1]  81 729  27
> 7^tep
[1]   2401 117649    343
> tep^(2+tep)
[1]    4096 1679616     243
> tep^(2+tep^3)
[1]  5.444518e+39 4.334835e+169  6.863038e+13
> rep(tep,10)
 [1] 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6
[24] 3 4 6 3 4 6 3
> rep(tep,l=31)
 [1] 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6
[24] 3 4 6 3 4 6 3 4
> #create a vector of the values of e^x cos(x)at x=4,4.1,4.2....7
> qap<-seq(4,7,by=0.1)
> exp(qap)*cos(qap)
 [1] -35.687732 -34.685042 -32.693695 -29.538816
 [5] -25.032529 -18.975233 -11.157417  -1.362099
 [9]  10.632038  25.046705  42.099201  61.996630
[13]  84.929067 111.061586 140.525075 173.405776
[17] 209.733494 249.468441 292.486707 338.564378
[21] 387.360340 438.397873 491.045163 544.494927
[25] 597.743415 649.569088 698.511407 742.850196
[29] 780.586182 809.423392 826.754210
> exp(qap)*sin(qap)
 [1]  -41.320016  -49.375076  -58.122190
 [4]  -67.521241  -77.508816  -87.994457
 [7]  -98.856669 -109.938735 -121.044378
[10] -131.933345 -142.316981 -151.853890
[13] -160.145806 -166.733804 -171.095016
[16] -172.640026 -170.711169 -164.581957
[19] -153.457895 -136.478991 -112.724257
[22]  -81.218568  -40.942224    9.156378
[25]   70.144561  143.085260  229.012558
[28]  328.902434  443.638334  573.971153
[31]  720.473289
> tanh(qap)
 [1] 0.9993293 0.9994508 0.9995504 0.9996319
 [5] 0.9996986 0.9997532 0.9997979 0.9998346
 [9] 0.9998646 0.9998891 0.9999092 0.9999257
[13] 0.9999391 0.9999502 0.9999592 0.9999666
[17] 0.9999727 0.9999776 0.9999817 0.9999850
[21] 0.9999877 0.9999899 0.9999918 0.9999933
[25] 0.9999945 0.9999955 0.9999963 0.9999970
[29] 0.9999975 0.9999980 0.9999983
> exp(tanh(qap))
 [1] 2.716459 2.716789 2.717060 2.717281 2.717463
 [6] 2.717611 2.717733 2.717832 2.717914 2.717980
[11] 2.718035 2.718080 2.718116 2.718146 2.718171
[16] 2.718191 2.718207 2.718221 2.718232 2.718241
[21] 2.718248 2.718254 2.718259 2.718263 2.718267
[26] 2.718270 2.718272 2.718274 2.718275 2.718276
[31] 2.718277
> #creat vectors
> #(0.1^3 0.2^1,0.1^6 0.2^4.....,0.1^36 0.2^34 )
> (0.1^seq(3,36,by=3))*(0.2^seq(1,34,by=3))
 [1] 2.000000e-04 1.600000e-09 1.280000e-14
 [4] 1.024000e-19 8.192000e-25 6.553600e-30
 [7] 5.242880e-35 4.194304e-40 3.355443e-45
[10] 2.684355e-50 2.147484e-55 1.717987e-60
> #(2,2^2/2,2^3/3,......,2^25/25)
> (2^(1:25))/(1:25)
 [1] 2.000000e+00 2.000000e+00 2.666667e+00
 [4] 4.000000e+00 6.400000e+00 1.066667e+01
 [7] 1.828571e+01 3.200000e+01 5.688889e+01
[10] 1.024000e+02 1.861818e+02 3.413333e+02
[13] 6.301538e+02 1.170286e+03 2.184533e+03
[16] 4.096000e+03 7.710118e+03 1.456356e+04
[19] 2.759411e+04 5.242880e+04 9.986438e+04
[22] 1.906502e+05 3.647221e+05 6.990507e+05
[25] 1.342177e+06
> #sum
> #
> #(i^3+4i^2) where i=10:100
> mohi<-10:100
> sum(mohi^3+4*mohi^2)
[1] 26852735
> rj<-1:25
> sum((2^rj)/rj+3*rj/(rj^2))
[1] 2807618
> #Function paste to create
> #("label 1","label 2".......,"label 30")
> paste("label",1:30)
 [1] "label 1"  "label 2"  "label 3"  "label 4" 
 [5] "label 5"  "label 6"  "label 7"  "label 8" 
 [9] "label 9"  "label 10" "label 11" "label 12"
[13] "label 13" "label 14" "label 15" "label 16"
[17] "label 17" "label 18" "label 19" "label 20"
[21] "label 21" "label 22" "label 23" "label 24"
[25] "label 25" "label 26" "label 27" "label 28"
[29] "label 29" "label 30"
> #("fn1","fn2",....,"fn30")
> paste("fn",1:30,sep="")
 [1] "fn1"  "fn2"  "fn3"  "fn4"  "fn5"  "fn6" 
 [7] "fn7"  "fn8"  "fn9"  "fn10" "fn11" "fn12"
[13] "fn13" "fn14" "fn15" "fn16" "fn17" "fn18"
[19] "fn19" "fn20" "fn21" "fn22" "fn23" "fn24"
[25] "fn25" "fn26" "fn27" "fn28" "fn29" "fn30"
> #Creat two vectors of random integer which are chosen with replacement from the integers 0,1,2.....,999. both vectors have lengh 250
> set.seed(50)
> rVec<-sample(0:999,250,replace = T)
> qVec<-sample(0:999,250,replace = T)
> #suppose x=(x1,x2,......xn) denote rVec and y=(y1,y2....yn) denot qVec
> # creat the vector (y2-x1,......,yn-xn-1)
> qVec[-1]-rVec[-length(rVec)]
  [1] -359  692 -724   40 -626 -719 -809  527  -89
 [10] -829  248  144 -749 -352 -220 -249  387 -492
 [19]   85 -106  303  -97 -436  146  282 -206 -385
 [28]  -96 -567 -757  287  277 -562  292  -89  -93
 [37] -847 -822 -203  679  309 -199 -273    4  -47
 [46]  142  122  414 -602 -304 -674   -8 -662 -168
 [55] -349  -63 -221  115    1 -600 -382 -487    2
 [64]  375   19 -113 -634  107   60   47  214 -325
 [73]  -49 -290  169  290 -624  457 -408  581 -189
 [82]  204  -80  409  209 -410  461   37 -127  185
 [91]  382 -446   44  -56 -270 -598 -378 -155  134
[100] -187  109  316 -139  158  305  -39 -119  182
[109]  441 -403 -107  615  614 -378 -464   31 -385
[118]  665  674 -217 -279 -406  -45 -489 -350 -451
[127]  -18  660  504   -6   60 -130 -379 -302 -219
[136]  -21  438  129 -201 -275  131  694  -96 -176
[145]  117 -113  887 -439 -126 -148  392 -158  444
[154] -291  232  -12 -274  477 -510  336 -759 -363
[163] -195 -220  160 -308 -333  302 -183  227  -12
[172]  428  665 -301   -8  222  -50 -444 -425 -650
[181] -424  318  154  238 -727   71  472  908  265
[190]  654 -644 -754  657 -382 -313  910 -381  394
[199] -596  602  397 -572  378 -274 -271  601 -791
[208] -378 -461   39  163 -118 -332 -170  -94  262
[217] -474  566 -273 -366 -400  374   42  100  135
[226]  609 -527  580 -219  128 -524  620 -206  410
[235] -280  -66  -50  252  279   48 -595  -59 -623
[244]  247  514   62 -102  475  287
> #Create the vector (sin(y1)/cos(x2),sin(y2)/cos(x3),......sin)
> Create the vector (sin(y1)/cos(x2),sin(y2)/cos(x3),......siny-1)

PROBABILITY

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