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I got this assignment from Generalized Linear Model class. At first glace it looked like it is an easy task, but there are a lot of subtle (at least in my opinion) things, which I would like to clarify with somebody. So here is the problem:

Assume that we are interested in random variable $Y=e^Z$. Variable $Z$ is obtained from simple linear regression i.e. $Z=a+bX+\varepsilon$, where $\varepsilon \sim N(0,4)$. Given 100 observations of $(Z,X)$ we have following empirical characteristics $\bar x=2$, $\sigma^2_x=1$, $\bar z = -1$ and $\operatorname {cov} (z,y)=-1 $.

  1. How do we construct approximate Wald's 95% confidence interval for parameter $\theta = E(Y|X=1/2)$, using the fact $E e^{\lambda U} = e^{a \lambda+\sigma^2\lambda^2/2}$, when $U \sim N(a, \sigma^2)$?

  2. How do we construct exact Wald's 95% confidence interval for individual prediction of $Y$ when $X=1/2$?

  3. Now assume that we do not know variance of $\varepsilon$ and from those 100 observations we know that the empirical variance of $Z$, $\sigma^2_z=4$.

I started by plugging in the empirical statistics to specify simple regression part and I got $Z=1-X+\varepsilon$, so the $$\operatorname{\log} Y =1-X+\varepsilon. $$ As the $\operatorname {log} Y$ is normaly distributed then the distribution of the mean of $Y$ should be lognormal? If $Y$ is log normal then I can't write write it in canonical exponential family form $$\operatorname{exp}\left\{ \frac{y \theta-b(\theta)}{a(\phi)}-c(y,\phi) \right\},$$ because I have $\operatorname{log}y$ instead of $y$ multiplied by $\theta$. That means that I do not know what $b(\theta)$ and other parameters are, which I need for standard error estimation.

So any tips on getting at least first question answered would be great!

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