Inferring Cross-Day Heterogeneity from Paired Signal Outcomes

Suppose each trading day has an unobserved hit probability $P \sim \mathrm{Beta}(\alpha,\beta)$. Conditional on $P$, two independent intraday signals $H_1$ and $H_2$ are Bernoulli$(P)$.

From data, you estimate $$E[H_1] = 0.60, \qquad P(H_1=1, H_2=1) = 0.42.$$ Use the method of moments to estimate $\alpha$ and $\beta$.