OCR FS1 AS 2017 Specimen — Question 6 13 marks

Exam BoardOCR
ModuleFS1 AS (Further Statistics 1 AS)
Year2017
SessionSpecimen
Marks13
TopicPoisson distribution
TypeExplain or apply conditions in context
DifficultyModerate -0.3 This is a straightforward Further Statistics question testing standard Poisson distribution knowledge: stating assumptions (textbook recall), applying the probability formula, and using additivity properties. While it requires multiple techniques including scaling parameters and combining distributions, these are routine FS1 procedures with no novel problem-solving or conceptual challenges beyond curriculum expectations.
Spec5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  1. State these two assumptions. [2]
  2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).
    1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
    2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
  2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
  3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

\begin{enumerate}[label=(\roman*)]
\item State these two assumptions. [2]

\item For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
\end{enumerate}

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution $\text{Po}(0.8)$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item \begin{enumerate}[label=(\alph*)]
\item Write down an expression for the probability that, in a given one minute period, exactly $r$ cars pass Sabrina's house. [1]

\item Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
\end{enumerate}

\item Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]

\item The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution $\text{Po}(1.5)$. Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR FS1 AS 2017 Q6 [13]}}
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