############################### # # # Targil 1 # # # ############################### # Question 1 gpa <- read.table("D:/Courses/regression/data sets/GPA.txt",header=F) x <- gpa[,2] y <- gpa[,3] S.XX <- sum((x-mean(x))^2) S.YY <- sum((y-mean(y))^2) S.XY <- sum((x-mean(x))*(y-mean(y))) b.1 <- S.XY / S.XX b.0 <- mean(y) - b.1 * mean(x) b.0 b.1 plot(x,y,xlab="Exceptance grade",ylab="Final grade") abline(b.0,b.1) b.0 + b.1 * 5 summary(lm(y~x)) # Question 2 z0 <- rnorm(300) z1 <- rnorm(300) z2 <- rnorm(300) x <- 175 + sqrt(30)*z1 + sqrt(70)*z0 y <- 175 + sqrt(30)*z2 + sqrt(70)*z0 plot(x,y,xlab="Fathers' height",ylab="Childrens height") abline(0,1,lty=2) abline(lm(y~x)) var(cbind(x,y)) # install 'mvtnorm' package from CRAN and load library(mvtnorm) mu <- c(175,175) sig <- cbind(c(100,70),c(70,100)) # 3D density plot x.ser <- seq(150,200,length=49) y.ser <- seq(150,200,length=49) dens.array <- array(dim=c(49,49)) for(i in 1:49) for(j in 1:49) dens.array[i,j] <- dmvnorm(c(x.ser[i],y.ser[j]),mu,sig) persp(x.ser,y.ser,dens.array) image(x.ser,y.ser,dens.array) contour(x.ser,y.ser,dens.array) # Conditional density plot x.ser[c(13,25,40)] plot(y.ser,dens.array[25,],type="l") lines(y.ser,dens.array[13,],type="l",col=3) lines(y.ser,dens.array[40,],type="l",col=4) plot(y.ser,dens.array[25,] / dnorm(x.ser[25],175,10),type="l") lines(y.ser,dens.array[13,]/ dnorm(x.ser[13],175,10),type="l",col=3) lines(y.ser,dens.array[40,]/ dnorm(x.ser[40],175,10),type="l",col=4) arrows(x.ser[25],0.1,x.ser[25],0,length=.1) arrows(x.ser[13],0.1,x.ser[13],0,length=.1,col=3) arrows(x.ser[40],0.1,x.ser[40],0,length=.1,col=4) # Sample observations xy.data <- rmvnorm(300,mu,sig) x <- xy.data[,1] y <- xy.data[,2] par(mfcol=c(1,2)) hist(x) hist(y) par(mfcol=c(1,1)) plot(x,y,xlab="Fathers height",ylab="Childrens height") abline(0,1) abline(lm(y~x),lty=2) var(cbind(x,y)) xy.data <- rmvnorm(100000,mu,sig) x <- xy.data[,1] y <- xy.data[,2] var(cbind(x,y))