Let $U_1$,...,$U_n$ be a random sample from the U(0,1)
a. Let $X$=-log($U$). Find the distribution of X
b. Let $Y$=$1/{\prod_{i=1}^n U_i^{1/n}}$, where $U_1$,...,$U_n$ be a random sample from the U(0,1) and n is very large. Find the approximate distribution of Y.
For part a, I use the following method, is it true?
$F(Y=y)$ = $P(Y<y)$ = $P(-logU<X)$ = $P(U>e^{-x})$ = $1-P(U<e^{-x})$ = $1-\int_0^{e^{-x}}dt$ = $1-e^{-x}$
Then, pdf=$e^{-x}$
For part b, i am not sure to use Central Limit Theorem & Delta method to do it
Your method for part a is correct.
For part b, realize that $$ \log Y = \frac1n\sum_{i=1}^n -\log(U_n) $$ We can now use the central limit theorem on $\log Y$, since $\log Y$ is an average of independent random variables each with the same distribution, namely, the distribution of $X$. Letting $\mu,\sigma^2$ be the mean and variance of $X$, the Central Limit Theorem says that $$ \sqrt{n}(\log(Y)-\mu)\to N(0,\sigma^2)\qquad\text{in distribution} $$ The above statement sets the stage for using the Delta method: for any continuous function $g$, where $g'(\mu)$ exists and $\neq 0$, the Delta method gives us $$ \sqrt{n}(g(\log Y)-g(\mu))\to N(0,g'(\mu)^2\sigma^2)\qquad\text{in distribution} $$ Using an appropriate choice of $g$, the above should let you determine the limiting distribution of $Y$.