Question

1. $X_1,\ldots,X_n \sim iid \sim N(0,1).$ Define $\bar{X}$ (sample mean) and $S^2=\sum_1^n X_i^2$ .

a) Show that $\bar{X}$ and $S^2$ are dependent.

b) Derive the conditional distribution of $\bar{X}$ , given $S^2$.

c) Determine c(u) such that P[$\bar{X}$ > c(u)|u] =alpha, where $\text{u}=S^2$.

Answer 1

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