This report is for first part of the course project of the Coursera course “Statistical Inference” which is a part of specialization “Data Science”. In this first part, we perform simulation exercises using exponential distributions. Simulations

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. For these simulations, we set lambda to 0.2. We investigate the distribution of averages of 40 exponentials. For this purpose, we perform a thousand or so simulated averages of 40 exponentials.

R code

Results

  1. Show where the distribution is centered at and compare it to the theoretical center of the distribution.
## [1] 4.971972
## [1] 5

Therefore, the center of distribution of averages of 40 exponentials is close to the theoretical center of the distribution.

  1. Show how variable it is and compare it to the theoretical variance of the distribution.
## [1] 0.7716456
## [1] 0.7905694

Therefore, the variability in distribution of averages of 40 exponentials is close to the theoretical variance of the distribution.

  1. Show that the distribution is approximately normal. plot of chunk unnamed-chunk-5

As evident from the Q-Q plot, the distribution of averages of 40 exponentials is very close to a normal distribution.

  1. Evaluate the coverage of the confidence interval for 1/lambda: \(\bar{X} \pm 1.96 \frac{s}{\sqrt{n}}\).
## [1] 3.459547 6.484397

Since, we consider the distribution of averages of exponentials, the standard deviation of this distribution already incorporates the \(\sqrt{n}\) term i.e. it is the standard error.

The confidence interval is given by [3.460, 6.484].