Question

Recall that \( \boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{n} \) are i.i.d. standard normal variables. Denote by \( A_{n} \) the sample mean of the squares of these variables: \[ A_{n}:=\overline{X_{n}^{2}}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}=\frac{1}{n}\left(X_{1}^{2}+X_{2}^{2}+\ldots+X_{n}^{2}\right) . \] \( 2(e) \) 20 points possible (graded, results hidden) Recal \( A_{a}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} \). Use the CLT and the fact that qo.o5 \( \left(A_{1}\right)=3.84 \) to approximate the 0.95 -quantile qo.05 \( \left(A_{m}\right) \) of the random variable \( A_{n} \) for sample sizes \( n=100 \) and \( n=10^{6} \). (Recall the \( 1-\alpha \) quantile \( q_{a}(Y) \) of a variable \( Y \) is defined by \( P\left(Y>q_{a}(Y)\right)=\alpha \) ) (Enter an answer accurate to at least 3 decimal places.) \( 90.05\left(A_{100}\right)= \) \( \square \) \( 9005\left(A_{10}\right)= \) \( \square \)

          Recall that \( \boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{n} \) are i.i.d. standard normal variables. Denote by \( A_{n} \) the sample mean of the squares of these variables:
\[
A_{n}:=\overline{X_{n}^{2}}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}=\frac{1}{n}\left(X_{1}^{2}+X_{2}^{2}+\ldots+X_{n}^{2}\right) .
\]
\( 2(e) \)
20 points possible (graded, results hidden)
Recal \( A_{a}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} \). Use the CLT and the fact that qo.o5 \( \left(A_{1}\right)=3.84 \) to approximate the 0.95 -quantile qo.05 \( \left(A_{m}\right) \) of the random variable \( A_{n} \) for sample sizes \( n=100 \) and \( n=10^{6} \).
(Recall the \( 1-\alpha \) quantile \( q_{a}(Y) \) of a variable \( Y \) is defined by \( P\left(Y>q_{a}(Y)\right)=\alpha \) )
(Enter an answer accurate to at least 3 decimal places.)
\( 90.05\left(A_{100}\right)= \) \( \square \)
\( 9005\left(A_{10}\right)= \) \( \square \)

Recall that X1, …, Xn are i.i.d. standard normal variables. Denote by An the sample mean of the squares of these variables:

An:=Xn^2=(1)/(n)∑i=1^n Xi^2=(1)/(n)(X1^2+X2^2+…+Xn^2) .

2(e)
20 points possible (graded, results hidden)
Recal Aa=(1)/(n)∑i=1^n Xi^2. Use the CLT and the fact that qo.o5 (A1)=3.84 to approximate the 0.95 -quantile qo.05 (Am) of the random variable An for sample sizes n=100 and n=10^6.
(Recall the 1-α quantile qa(Y) of a variable Y is defined by P(Y>qa(Y))=α )
(Enter an answer accurate to at least 3 decimal places.)
90.05(A100)= □
9005(A10)= □

Added by Kevin C.

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