Standard +0.3 This is a straightforward application of geometric distribution expectation after calculating a Poisson probability. Students must recognize the Poisson setup (λ=4 for 20 minutes), find P(Y≥8), then apply E(X)=1/p. The multi-step nature and context make it slightly above average, but the required techniques are standard for Further Statistics.
2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, \(X\), of nights on which I look for shooting stars, up to and including the first successful night.
Find \(\mathrm { E } ( X )\).
2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.\\
From tomorrow, I will count the number, $X$, of nights on which I look for shooting stars, up to and including the first successful night.
Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{OCR Further Statistics 2018 Q2 [7]}}