1. X1,…,Xn∼iid∼N(0,1).X_1,\ldots,X_n \sim iid \sim N(0,1).X1,…,Xn∼iid∼N(0,1). Define Xˉ\bar{X}Xˉ (sample mean) and S2=∑1nXi2S^2=\sum_1^n X_i^2S2=∑1nXi2 .
a) Show that Xˉ\bar{X}Xˉ and S2S^2S2 are dependent.
b) Derive the conditional distribution of Xˉ\bar{X}Xˉ , given S2S^2S2.
c) Determine c(u) such that P[Xˉ\bar{X}Xˉ > c(u)|u] =alpha, where u=S2\text{u}=S^2u=S2.
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