A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for the following situations, showing the pattern that can be expected, which would indicate a departure from the assumed model.
1) The true model is $\sqrt{Y_{i}} = x_i^{'}\beta + \epsilon_i$.
2) The true model is $\frac{1}{Y_{i}} = x_i^{'}\beta + \epsilon_i$
3) $Y_i$ follows a chi-squared distribution with 4 degrees of freedom.
4) $\epsilon_{i} + \frac{1}{3}$ follows a gamma distribution with $\alpha=4, \beta=12$. Recall that the expectation of a gamma distribution is $\frac{\alpha}{\beta}$.
I wasn't able to solve the first three, but for the last one, I sketched the PDF of the Gamma distribution, realised that it has heavy tails/high kurtosis (I think), and so plotted the relevant residual plot for a distribution with heavy tails. Is this correct?
How would one go about solving the first three? If anyone can provide a solid reasoning which I can use to tackle different questions of similar form, that would be much appreciated.