Statistical Inference Part 1
Hazim Hanif
26 September 2016
Project Description
In this project you will investigate the exponential distribution in R and compare it with the Central Limit Theorem. The exponential distribution can be simulated in R with rexp(n, lambda) where lambda is the rate parameter. The mean of exponential distribution is 1/lambda and the standard deviation is also 1/lambda. Set lambda = 0.2 for all of the simulations. You will investigate the distribution of averages of 40 exponentials. Note that you will need to do a thousand simulations.
Setting the parameter
n <- 40
lambda <- 0.2
simulation <- 1000Calculation of theoretical values
mean_t = 1/lambda
sd_t = ((1/lambda) * (1/sqrt(n)))
var_t = sd_t^2Calculation of actual values
data <- matrix(rexp(n*simulation, lambda), simulation)
row_means <- apply(data,1,mean)
mean_act <- mean(row_means)
sd_act <- sd(row_means)
var_act <- var(row_means)Question
1. Show where the distribution is centered at and compare it to the theoretical center of the distribution.
The actual distribution is centered at 5.0450596 while the theoretical distribution is centered at 5
2. Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution.
- The actual variance is 0.6541736 while the theoretical variance is 0.625
- The actual standard deviation is 0.80881 while the theoretical standard deviation is 0.7905694
| Variable | Theoretical Value | Actual Value |
|---|---|---|
| Mean | 5.0450596 | 5 |
| Standard Deviation | 0.80881 | 0.7905694 |
| Variance | 0.6541736 | 0.625 |
3. Show that the distribution is approximately normal.
dfrow_means<-data.frame(row_means)
pp<-ggplot(dfrow_means,aes(x=dfrow_means))
pp<-pp+geom_histogram(binwidth = lambda,fill="pink",color="black",aes(y = ..density..))
pp<-pp + labs(title="Density of 40 Numbers from Exponential Distribution", x="Mean of 40 Selections", y="Density")
pp<-pp + geom_vline(xintercept=mean_act,size=1.0, color="black") # actual mean line
pp<-pp + stat_function(fun=dnorm,args=list(mean=mean_act, sd=sd_act),color = "blue", size = 1.0)
pp<-pp + geom_vline(xintercept=mean_t,size=1.0,color="yellow",linetype = "longdash")
pp<-pp + stat_function(fun=dnorm,args=list(mean=mean_t, sd=sd_t),color = "green", size = 1.0)
pp## Don't know how to automatically pick scale for object of type data.frame. Defaulting to continuous.
- Black line is the actual mean
- Yellow dotted line is the theoretical mean
- Green curve is the theoretical standard deviation
- Blue curve is the actual standard deviation
The plot shows that Central Limit Theory works by trying to shape the actual data to follow the normal curve.