Question

The weak law of large numbers states that the mean of a sample is a consistent estimator of the mean of the population. That is, as we increase the sample size, the mean of the sample converges in probability to the expected value of the distribution that the data comes from, provided that expected value is finite. Consider a numerical example, a Student t distribution with n = 5, the same as we have already seen earlier in this module. As we increase the sample size, the sample mean converges to the expected value of the Student t distribution,

R code :

sampleSize<-2^seq(2, 20)
mean<-rep(0, length(sampleSize))
for(i in 1:length(sampleSize)) {
mean[i]<-mean(rt(sampleSize[i],5))
}
plot(log(sampleSize), mean)

Question 15: What is the expected value of the Student t distribution?

The law of large numbers only applies if the distribution that the data comes from has a finite expected value. In the previous section of this module, we saw a distribution with an infinite variance. There are distributions with infinite or undefined expected values as well. lim | | 1 ( n ) n P Y q e ®¥ - < = Here is an the same example with a Cauchy distribution instead of Student t distribution:

sampleSize<-2^seq(2, 20)
mean<-rep(0, length(sampleSize))
for(i in 1:length(sampleSize)) {
mean[i]<-mean(rcauchy(sampleSize[i], 0, 1))
}
plot(log(sampleSize), mean)

There is no convergence.

Question 16: As you see from the above, the weak law of large numbers doesn't apply to random variables obtained from a Couchy distribution. Does the central limit theorem (it's most common version) apply to i.i.d. variables that have the Couchy distribution? - yes - no

Answer 1

15) If the degrees of freedom of the t-distribution is greater than 1, then the mean is 0. Otherwise, for degrees of freedom equal to 1, the mean is undefined.

16) No it doesn't apply to Cauchy distribution. First of all the mean and variance of a Cauchy distribution are undefined. Recall in CLT that given iid random variables , with mean and a finite variance then:

        

And we have that:                  

But, for the Cauchy distribution, neither of the two parameters exist.