A-Level Mathematics May/June 2024 Q1(a): A random variable X has the distribution Po(145). (a) Use a suitable approximating dist…

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  The random variable $X$ has the distribution $Po(145)$.

  Since the parameter $\lambda = 145$ is large ($\lambda > 10$), the Poisson distribution can be approximated by a Normal distribution, $Y$, with mean $\mu = \lambda$ and variance $\sigma^2 = \lambda$.

  So, $X \sim Po(145)$ is approximated by $Y \sim N(145, 145)$.

  We want to calculate $P(X \le 150)$. We must apply a continuity correction when moving from the discrete Poisson distribution to the continuous Normal distribution.

  $P(X \le 150) \approx P(Y < 150.5)$

  Now, we standardise the variable to find the Z-score:

  $Z = \frac{Y - \mu}{\sigma} = \frac{Y - 145}{\sqrt{145}}$

  $P(Y < 150.5) = P\left(Z < \frac{150.5 - 145}{\sqrt{145}}\right)$

  $P\left(Z < \frac{5.5}{\sqrt{145}}\right) = P(Z < 0.45675...)$

  Using the standard normal distribution tables or a calculator, we find the probability corresponding to this Z-value.

  $\Phi(0.45675...) = 0.6761...$

  Rounding to 3 significant figures, the final answer is:

  $P(X \le 150) \approx 0.676$

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  How the marks are awarded

  • B1 — For stating or implying the use of the Normal approximation with mean $\mu = 145$ and variance $\sigma^2 = 145$.
  • M1 — For applying the standardisation formula, including the correct continuity correction to get 150.5. The expression $\frac{150.5 - 145}{\sqrt{145}}$ earns this mark.
  • M1 — For finding the correct probability area from the Normal distribution tables or a calculator, corresponding to their calculated z-value. This is shown by finding $\Phi(0.457)$.
  • A1 — For the correct final answer of 0.676 (or more accurately 0.6761) correctly rounded to 3 significant figures.

  Common mistakes

  • Forgetting the continuity correction: Using 150 instead of 150.5 in the standardisation formula. This leads to a z-score of $\frac{150-145}{\sqrt{145}} \approx 0.415$ and an incorrect final answer of 0.661.
  • Applying the wrong continuity correction: Using 149.5 instead of 150.5. This would be the correction for $P(X < 150)$, not $P(X \le 150)$, and leads to an incorrect answer of 0.646.
  • Using variance instead of standard deviation: Dividing by 145 instead of $\sqrt{145}$ in the standardisation formula. This results in a very small z-value and a probability close to 0.5.
  • Premature rounding: Rounding the z-value (0.45675...) too early, for example to 0.46, before looking up the probability, which can lead to an inaccurate final answer and loss of the A1 mark.

  Examiner tip: When approximating a discrete distribution (like Poisson or Binomial) with a continuous one (Normal), always remember to apply a continuity correction to account for the change from integer values to a continuous scale.

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