Question

A friend is behind the computer generating values from X∼Exp(λ), but won’t tell you whatλis. She will, however, tell you 35 different random values from the distribution,X1 through X35.

1. Explain what the population and parameter are for this setting.

2. Based on the data your friend gives, you decide your best guess forλshould be 1/X, where X=(X1+···+X35)/35 is the average of the 35 numbers. Explain why this is a reasonable statistic to use based on your knowledge of the exponential distribution

3.For a particular set of 35 random values, your statistic equals some number (hopefully nearλ). As you explore 35 new values, you get a (potentially) different value for your statistic. The sampling distribution shows all the possible values for the statistic and how likely they are. Create a rough picture of the sampling distribution for 1/X by generating 10000 different sets of 35 numbers,and for each set, finding 1/X. Use Exp(λ= 13) when generating numbers. Include your code and a sketch of the sampling distribution.

4.What shape does the sampling distribution appear to have and where does it appear to be centered?

Answer 1

1) The population is exponentially distributed with rate = lambda

2) We know that when Xi ~ exp(lambda)

then the MLE (maximum likelihood estimator) of lambda is 1/X

where, X = (X1 + X2 + .... + X35)/35

3)

#### R Statistical Software ####
n = 35
estimates = c()
for(i in 1:10000){
sample = rexp(n, rate = 13)
lambda_hat = 1/mean(sample)
estimates = c(estimates, lambda_hat)
}
hist(estimates)

4) the distribution is positively skewed, it is centred around 13.