In this problem, we perform a simulation study about clustered binary outcomes with sample size 1000 .

(a) Generate random variable \(X\) from \(\mathrm{N}(0,1)\).

(b) For each \(X\), generate five binary responses with response probability \(\frac{\exp (x)}{1+\exp (x)}\) in two scenarios and sum up the five binary to obtain the response variable \(Y\). In the first, all the five binary responses are independent. In the second, all the five binary responses are positively correlated with the same correlation.