Question

Do the following in the program R

Suppose a random variable X follows the Rayleigh distribution with probability density function given by f(x) = x/sigma^2 e^, 0 lessthanorequalto x < infinity, 0 < sigma < infinity. The cumulative distribution function is F(x) = 1 - e^, 0 lessthanorequalto x < infinity. The mean and variance of a Rayleigh random variable are. respectively. E(X) = sigma squareroot pi/2 and var(X) = (4 - pi/2) sigma^2. Plot the Rayleigh probability density function f(x) for sigma = 0.5, 1.0, 2, 3.0. and 4.0. Your graphs must be in the same Figure including (i) different colors, (ii) different line types, and (iii) math expression in the title and legend. Plot the Rayleigh cumulative distribution function F(x) for sigma = 0.5, 1.0, 2, 3.0, and 4.0. Your graphs must be in the same Figure including: (i) different colors, (ii) different line types, and (iii) math expression in the title and legend.

Answer 1

a)

Program screenhot of probability density function:

Sample output screenshot:

Code to copy:

set.seed(42)

# Calculate comulative distribution function

ecdf1 <- ecdf(rnorm(100)*0.5)

ecdf2 <- ecdf(rnorm(100)*1.0)

ecdf3 <- ecdf(rnorm(100)*2.0)

ecdf4 <- ecdf(rnorm(100)*3.0)

ecdf5 <- ecdf(rnorm(100)*4.0)

#attr(ecdf3,"call") <- "Cumulative Distribution Function"

plot(ecdf3, verticals=TRUE, do.points=FALSE, col='green')

plot(ecdf2, verticals=TRUE, do.points=FALSE, add=TRUE)

plot(ecdf1, verticals=TRUE, do.points=FALSE, add=TRUE, col='blue')

plot(ecdf5, verticals=TRUE, do.points=FALSE, add=TRUE, col='pink')

plot(ecdf4, verticals=TRUE, do.points=FALSE, add=TRUE, col='red')

b)

Program screenshot of comulative distribution function:

Sample output screenshot:

Code to copy:

set.seed(4000)

xseq<-seq(-8,10,.1)

par(mfrow=c(1,1), mar=c(3,4,4,3))

# Calculate probality density values

value1<-dnorm(xseq, 0,0.5)

value2<-dnorm(xseq, 0,1.0)

value3<-dnorm(xseq, 0,2.0)

value4<-dnorm(xseq, 0,3.0)

value5<-dnorm(xseq, 0,4.0)

# Display values in a grap using plot method

plot(xseq, value1, col="blue",xlab="", ylab="Density",main="Probability Density Fucntion is dnorm()" ,type="l",lwd=2, cex=2,cex.axis=.8)

# The method lines() is used ot attach the lines in a grap

lines(xseq, value2, col="black",xlab="", ylab="Density", type="l",lwd=2, cex=2,cex.axis=.8)

lines(xseq, value3, col="green",xlab="", ylab="Density", type="l",lwd=2, cex=2,cex.axis=.8)

lines(xseq, value4, col="red",xlab="", ylab="Density", type="l",lwd=2, cex=2,cex.axis=.8)

lines(xseq, value5, col="purple",xlab="", ylab="Density", type="l",lwd=2, cex=2,cex.axis=.8)