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1) Let to > 0 and X be a random variable such that E(etX) < co for all t R with It| < to. Let
1) Let to > 0 and X be a random variable such that E(etX) < co for all t R with It| < to. Let
1(t)=1og(E(etX)) for ltl < to. From one of our
theorems in class, we know that y has derivatives of all orders, so you may use this in your solutions.
- What is W(0)?
- Write W'(0) in terms of the moments of X.
- Write 4" (0) in terms of the moments of X.
- Write l' (0) in terms of the moments of X.
- Write out the first four terms (i.e. up terms of degree 3) in the Taylor series for y (based at 0).