5.02i Poisson distribution: random events model

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OCR Further Statistics 2017 Specimen Q5
8 marks Standard +0.3
The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).
  1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. [2]
Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).
    1. Write down an expression for P\((X = r)\). [1]
    2. Hence find P\((X = 3)\). [1]
  2. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match. [4]
OCR FS1 AS 2017 Specimen Q6
13 marks Moderate -0.3
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]
Pre-U Pre-U 9795/2 2014 June Q5
13 marks Standard +0.3
  1. The discrete random variable \(X\) has a Poisson distribution with mean \(\lambda\). Use the probability generating function for \(X\) to show that both the mean and the variance have the value \(\lambda\). [5]
  2. The number of eggs laid by a certain insect has a Poisson distribution with variance 250. Find, using a suitable approximation, the probability that between 230 and 260 (inclusive) eggs are laid. [5]
  3. An insect lays 250 eggs. The probability that any egg that is laid survives to maturity is 0.1. Use a suitable approximation to find the probability that more than 30 eggs survive to maturity. [3]
Pre-U Pre-U 9795/2 Specimen Q9
10 marks Standard +0.3
A certain type of fossil occurs at a mean rate of \(0.5\) per square metre at a particular location.
  1. State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
  2. Find the probability of at least 7 of these fossils occurring in an area of \(10 \text{ m}^2\). [2]
  3. Given that at least 4 such fossils have occurred in an area of \(5 \text{ m}^2\), find the probability that there will be more than 6 found in this area of \(5 \text{ m}^2\). [3]
  4. Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than \(0.999\). Give your answer to the nearest square metre. [4]