The CLT provides an approximate sampling distribution for the arithmetic average Y Ì„ of a iid
random sample Y1,...,Yn ∼ f(y). The parameters of the approximate sampling distribution depend on the mean and variance of the underlying random variables (i.e., the population mean and variance). The approximation can be written to emphasize this, using the expec- tation and variance of one of the random variables in the sample instead of the parameters μ, σ2:
̄· varYi Y∼N EYi, n
For the following population distributions f, write the approximate distribution of the sample mean.
(a) Exponentialwithrateβ: f(y)=βexp{−βy} (b) Chi-square with ν degrees of freedom: f(y) =
(c) Poisson with rate λ: P (Y = y) = λy exp{−λ} y!
1 ν yν2 exp−y Γ ( ν2 ) 2 2 2