Breadth in a factor model

Stocks follow $r_i=\beta_if+\varepsilon_i$ with idiosyncratic volatility 1.5% for every stock. (a) With $\beta_i=1$ for all stocks, compute the expected fraction of advancers when the factor return is $+1\%$, $0$ and $-1\%$. (b) Half the stocks have $\beta=1.6$ and half $\beta=0.4$; recompute for $f=+1\%$. (c) Construct an example in which the cap-weighted index rises while fewer than half the stocks advance, and say what the divergence measures.