Show that if E(X|Y = y) = c for some constant c, then X and Y are uncorrelated ? This question is to help you understand the idea of a sampling distribution. Let X1,…,Xn be iid with mean µ and variance σ2. Let
is a statistic, that is, a function of the data. Since
is a random variable, it has a distribution. This distribution is called the sampling distribution of the statistic. Recall from Theorem 3.17 that
Don’t confuse the distribution of the data fX and the distribution of the statistic
To make this clear, let X1,…,Xn ∼ Uniform(0, 1). Let fX be the density of the Uniform(0, 1). Plot fX. Now let
Plot them as a function of n. Interpret. Now simulate the distribution of Plot them as a function of n. Interpret. Now simulate the distribution of for n = 1, 5, 25, 100. Check that the simulated values agree with your theoretical calculations. What do you notice about the sampling distribution of as n increases?