[SOLVED] CS ##

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set.seed(869)
## 1 (fair) coin-flip:
results <- sample(x = c(“H”, “T”), size = 1)## Relative frequency of “H” in 1 coin-flipslength(results[results==”H”])/1## 10 (fair) coin-flips:results <- sample(x = c(“H”, “T”), size = 10, replace = TRUE)## Relative frequency of “H” in 10 coin-flipslength(results[results==”H”])/10## 100000 (fair) coin-flips:results <- sample(x = c(“H”, “T”), size = 100000, replace = T)## Relative frequency of “H” in 100000 coin-flipslength(results[results==”H”])/100000## —- fig.align=”center”—————————————————-library(“bivariate”)F <- nbvcdf(mean.X = 0, mean.Y = 0, sd.X = 1, sd.Y = 1, cor = .5)## Plotpar(mfrow=c(1,1), family=”serif”)plot(F)## —- fig.align=”center”—————————————————-## Install the package if not installed yet# install.packages(“bivariate”)## Load the packagelibrary(“bivariate”)## The vignette describes the package very well:# vignette(“bivariate”) f <- nbvpdf(mean.X = 0, mean.Y = 0, sd.X = 1, sd.Y = 1, cor = .5)## Plotpar(family=”serif”)plot(f)## —- fig.align=”center”, echo=FALSE, fig.width=5, fig.height=5————-library(“scales”)par(lwd=2,mgp=c(1,1,0), family=”serif”)# Modified to extract diagonal.bivariate.normal <- function(x, mu, Sigma) exp(-.5 * diag(t(x-mu) %*% solve(Sigma) %*% (x-mu))) / sqrt(2 * pi * det(Sigma))mu<- c(0,0)Sigma <- matrix(c(1,.8,.8,1), nrow=2)x1 <- seq(-3, 3, length.out=50)x2 <- seq(-3, 3, length.out=50)# plot(1:10,axes=FALSE, frame.plot=TRUE, lwd=1)# z can now be calculated much easier.z <- bivariate.normal(t(expand.grid(x1,x2)),mu,Sigma)dim(z)<-c(length(x1),length(x2))contour(x1, x2, z, col=”#4545FF”, drawlabels=FALSE, nlevels=5,xlab=expression(X[1]), ylab=expression(X[2]),lwd=1,xlim=c(-3,4),ylim=range(x2),frame.plot=FALSE,axes=FALSE,xaxs = “i”, yaxs = “i”)image(x1, x2, z, col=gray(seq(1,.2,len=10)), add=TRUE)contour(x1, x2, z, col=alpha(“darkblue”, 0.5), drawlabels=FALSE, nlevels=5, add=TRUE)#box(lwd=.5)axis(1,at = -3:3, labels=FALSE,lwd.ticks=1)axis(2,labels=FALSE,lwd.ticks=1)# Marginal probability distribution: http://mpdc.mae.cornell.edu/Courses/MAE714/biv-normal.pdf# Please check this, I’m not sure it is correct.marginal.x1<-function(x)exp((-(x-mu[1])^2)/2*(Sigma[1,2]^2)) / (Sigma[1,2]*sqrt(2*pi))marginal.x2<-function(x)exp((-(x-mu[1])^2)/2*(Sigma[2,1]^2)) / (Sigma[2,1]*sqrt(2*pi))# Left side solidx.s<- seq(from=min(x1),to=max(x1),by=0.1)vals<-marginal.x2(x.s)lines(vals-abs(min(x1)),x.s,lty=1,lwd=1, col=”darkred”)# Bottom side solidvals<-marginal.x1(x.s)lines(x.s,vals-abs(min(x2)),lty=1,lwd=1, col=”darkorange”)legend(“topleft”, col=c(“darkred”,”darkorange”), lty=1, legend = c(expression(f[1]),expression(f[2])), bty=”n”, title = “Marginal Densities”)legend(“topright”, col=c(“darkblue”), lty=1, legend = expression(f(x[1],x[2])), bty=”n”, title = “Joint Density”)## —- fig.align=”center”, echo=FALSE, fig.width=5, fig.height=5————-par(lwd=2,mgp=c(1,1,0), family=”serif”)# Modified to extract diagonal.bivariate.normal <- function(x, mu, Sigma) exp(-.5 * diag(t(x-mu) %*% solve(Sigma) %*% (x-mu))) / sqrt(2 * pi * det(Sigma))mu<- c(0,0)Sigma <- matrix(c(1,.8,.8,1), nrow=2)x1 <- seq(-3, 3, length.out=50)x2 <- seq(-3, 3, length.out=50)# plot(1:10,axes=FALSE, frame.plot=TRUE, lwd=1)# z can now be calculated much easier.# z <- bivariate.normal(t(expand.grid(x1,x2)),mu,Sigma)# dim(z)<-c(length(x1),length(x2))# contour(x1, x2, z, col=”#4545FF”, drawlabels=FALSE, nlevels=5,# xlab=expression(X[1]), ylab=expression(X[2]),lwd=1,xlim=c(-3,4),ylim=range(x2),frame.plot=FALSE,axes=FALSE,xaxs = “i”, yaxs = “i”)z <- bivariate.normal(t(expand.grid(x1,x2)),mu,Sigma)dim(z)<-c(length(x1),length(x2))contour(x1, x2, z, col=”#4545FF”, drawlabels=FALSE, nlevels=5,xlab=expression(X[1]), ylab=expression(X[2]),lwd=1,xlim=c(-3,4),ylim=range(x2),frame.plot=FALSE,axes=FALSE,xaxs = “i”, yaxs = “i”)image(x1, x2, z, col=gray(seq(1,.2,len=10)), add=TRUE)contour(x1, x2, z, col=alpha(“darkblue”, 0.5), drawlabels=FALSE, nlevels=5, add=TRUE)#box(lwd=.5)axis(1,labels=FALSE,lwd.ticks=1)axis(2,labels=FALSE,lwd.ticks=1)abline(v=.7, col=1, lwd=1.1, lty=2)text(2, -2, labels=expression(x[1]==0.7))# Dottedf<- function(x1,x2) bivariate.normal(t(cbind(x1,x2)),mu,Sigma)x.s<- seq(from=min(x1),to=max(x1),by=0.1)vals <- f(x1=0.7,x2=x.s)lines(vals-abs(min(x1)),x.s,lty=2,lwd=1.2, col=”darkgreen”)marginal.x1<-function(x)exp((-(x-mu[1])^2)/2*(Sigma[1,2]^2)) / (Sigma[1,2]*sqrt(2*pi))marginal.x2<-function(x)exp((-(x-mu[1])^2)/2*(Sigma[2,1]^2)) / (Sigma[2,1]*sqrt(2*pi))# Left side solidvals<-marginal.x2(x.s)lines(vals-abs(min(x1)),x.s,lty=1,lwd=1, col=gray(.5, 0.5)) # Bottom side solidvals<-marginal.x1(x.s)lines(x.s,vals-abs(min(x2)),lty=1,lwd=1, col=gray(.5, 0.5))# legend(“topright”, col=c(“darkred”,”darkorange”), lty=1, legend = c(expression(f[1]),expression(f[2])), bty=”n”,#title = “Marginal Densities”)legend(“topleft”, col=”darkgreen”, lty=2.1, lwd=.98, legend = expression(f(x[2]~ “|” ~ x[1]==0.7)), bty=”n”, title = “Conditional Density”)## —————————————————————————set.seed(51)## Set the parameter kk <- 10## Draw one realization from the discrete uniform distributionsample(x = 1:k, size = 1, replace = TRUE)## —————————————————————————set.seed(51)## Set the parameter pp <- 0.25## Draw n realization from the discrete uniform distributionn <- 5sample(x = c(0,1), size = n, prob = c(1-p, p), replace=TRUE)## Alternatively:## (Bernoulli(p) equals Binomial(1,p))rbinom(n = n, size = 1, prob = p)## —————————————————————————set.seed(51)## Set the parameters n and psize <- 10 # number of trialsp<- 0.25 # prob of success## Draw n realization from the binomial distribution:n <- 5rbinom(n = n, size = size, prob = p)## —————————————————————————## Drawing from the uniform distribution:n <- 10a <- 0b <- 1runif(n = n, min = a, max = b)## —- echo = T, eval = T, message = F, warning = F, fig.align=’center’——# draw a plot of the N(0,1) PDFcurve(dnorm(x),xlim = c(-3.5, 3.5),ylab = “Density”, main = “Standard Normal Density Function”) ## —————————————————————————## Drawing from the uniform distribution:n <- 12mu<- 0sigma <- 1rnorm(n = n, mean = mu, sd = sigma)## —- echo = T, eval = T, message = F, warning = F, fig.align=’center’——# plot the PDFcurve(dchisq(x, df = 3), xlim = c(0, 10), ylim = c(0, 1), col = “blue”,ylab = “”,main = “pdf and cdf of Chi-Squared Distribution, M = 3”)# add the CDF to the plotcurve(pchisq(x, df = 3), xlim = c(0, 10), add = TRUE, col = “red”)# add a legend to the plotlegend(“topleft”,c(“PDF”, “CDF”),col = c(“blue”, “red”),lty = c(1, 1))## —- echo = T, eval = T, message = F, warning = F, fig.align=’center’——# plot the density for M=1curve(dchisq(x, df = 1), xlim = c(0, 15), xlab = “x”, ylab = “Density”, main = “Chi-Square Distributed Random Variables”)# add densities for M=2,…,7 to the plot using a ‘for()’ loop for (M in 2:7) {curve(dchisq(x, df = M),xlim = c(0, 15), add = T, col = M)}# add a legendlegend(“topright”,as.character(1:7),col = 1:7 ,lty = 1,title = “D.F.”)## —- echo = T, eval = T, message = F, warning = F, fig.align=’center’——# plot the standard normal densitycurve(dnorm(x), xlim = c(-4, 4), xlab = “x”, lty = 2, ylab = “Density”, main = “Densities of t Distributions”)# plot the t density for M=2curve(dt(x, df = 2), xlim = c(-4, 4), col = 2, add = T)# plot the t density for M=4curve(dt(x, df = 4), xlim = c(-4, 4), col = 3, add = T)# plot the t density for M=25curve(dt(x, df = 25), xlim = c(-4, 4), col = 4, add = T)# add a legendlegend(“topright”,c(“N(0, 1)”, “M=2”, “M=4”, “M=25”),col = 1:4,lty = c(2, 1, 1, 1))## —- echo = T, eval = T, message = F, warning = F————————–pf(2, df1 = 3, df2 = 14, lower.tail = F)## —- echo = T, eval = T, message = F, warning = F, fig.align=’center’——# define coordinate vectors for vertices of the polygonx <- c(2, seq(2, 10, 0.01), 10)y <- c(0, df(seq(2, 10, 0.01), 3, 14), 0)# draw density of F_{3, 14}curve(df(x ,3 ,14), ylim = c(0, 0.8), xlim = c(0, 10), ylab = “Density”,main = “Density Function”)# draw the polygonpolygon(x, y, col = “orange”)