SIprojectPart1

Statistical Inference Course Project - Part 1

Overview:

The project report covers several series of exponential distribution iterations, and the result set is compared to a normal distribution and the theoretical mean and variance.

This report covers the points listed below for the problem statement:

Illustrate via simulation and associated explanatory text the properties of the distribution of the mean of 40 exponentials. You should:
1. Show the sample mean and compare it to the theoretical mean of the distribution.
2. Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution.
3. Show that the distribution is approximately normal.

Simulations:

The simulation can be built in R using the rexp(n,l) function, where n is the number of exponentials per set, and l, or lambda is the rate (a constant of these purposes of 0.2). We need to build a single column data frame with the average of each of the sets. We iterate 1k times, or 1k simulations, or s, and use the mean function to calculate the average. The code is as follows:

set.seed(123456789);
l<-0.2        # lambda
n<-40         # number of exponentials
s<-1000       # number of simulations

# initialize a data frame with a row count of the number of simulations
df<-data.frame(mean=numeric(s))

# iterate 1 to the number of simulations variable
for (i in 1:s) {
  ss<-rexp(n,l)     # simulation set of n exponential with l lambda
  df[i,1]<-mean(ss) # mean of simulation set
}

Sample Mean versus Theoretical Mean:

With our sample data in place, we will compare the sample mean to the theoretical mean. The latter is defined as 1/lambda; we will define tm in this manner.

tm<-1/l             # theoretical mean
tm

## [1] 5

We can define the sample mean as the average of the interated set; am is defined in the following manner.

am<-mean(df$mean)   # actual mean
am

## [1] 5.005439

As you can see from the above two results, the sample mean (or actual mean) of 5.1 is very close to the theoretical mean of 5.0. When we plot the sample set below and place a vertical line on both mean results, the histograms look almost identical.

par(mfrow=c(1,2))
hist(df$mean,probability=T,main=paste('Theoretical Mean of ',format(round(tm,2),nsmall=2)),ylim=c(0,0.55),col='gray',xlab='Means of Simulation Set')
abline(v=tm,col='blue',lwd=5)
hist(df$mean,probability=T,main=paste('Actual Mean of ',format(round(am,2),nsmall=2)),ylim=c(0,0.55),col='gray',xlab='Means of Simulation Set')
abline(v=am,col='yellow',lwd=5)

Sample Variance versus Theoretical Variance:

We can futher compare by defining the theoretical variance as 1/lambda squared, divided by the number of exponential observations, or n. We will define tv in this way as seen below.

tv<-((1/l)^2)/n     # theoretical variance
tv

## [1] 0.625

We can use the R var function to determine the actual variance of the observations. We can define av as follows.

av<-var(df$mean)    # actual variance
av

## [1] 0.6010095

The results of the theoretical variance as compared to actual variance are very close, as noted above. So, the sample data is ‘spread-out’ in a similar fashion to what is expected.

Distribution:

Finally, if we plotted the a normal curve and overlayed to the sample set along with the actual distribution curve, we can see very clearly that they are very similar; in turn, the sample set can be defined as normal, in terms of distribution.

par(mfrow=c(1,1))
hist(scale(df$mean),probability=T,main='',ylim=c(0,0.5),xlab='')
curve(dnorm(x,0,1),-3,3, col='red',add=T) # normal distribution
lines(density(scale(df$mean)),col='blue') # actual distribution 
legend(2,0.4,c('Normal','Actual'),cex=0.8,col=c('red','blue'),lty=1)