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CIE A-Level Maths Study Notes

5.1.5 Normal Approximation to the Poisson Distribution

The approximation of the Poisson distribution using a normal distribution is an invaluable tool in statistics, particularly beneficial when dealing with large data sets. This approximation simplifies complex statistical problems, making them more approachable and understandable.

Introduction

  • When the mean (λ)(λ) of a Poisson distribution is large (usually λ≥15), using a normal distribution for approximation is effective.
  • This method simplifies calculations for large λ values.

Normal Approximation Criteria

  • Applicable when λ is large (≥15).
  • As λ increases, the Poisson distribution looks more like a normal distribution.

Continuity Correction

  • Needed because Poisson is discrete and normal distribution is continuous.
  • Adjust the range of values slightly for a more accurate approximation.

Examples

Example 1: Call Centre Probability

A call centre receives an average of 18 calls per hour. Find the probability of receiving at most 20 calls in an hour.

Solution:

  • Given: λ=18,X20\lambda = 18, X \leq 20.
  • Applying Continuity Correction: Adjust X to 20.5.
  • Converting to Z-Score:
    • $Z = \frac{X - λ}{√λ} </li></ul></li><li><strong>Calculate:</strong></li></ul></li><li><strong>Calculate: </strong>Z = \frac{20.5 - 18}{\sqrt{18}} \approx 0.59.</li><li><strong>Result:</strong>Probabilityfor</li><li><strong>Result: </strong>Probability for Z = 0.59is 72.24 is ~72.24%. So, 72.24% chance of \leq 20calls.</li></ul><imgsrc="https://tutorchaseproduction.s3.euwest2.amazonaws.com/8edee42e1233401cbab538e832b7c69cfile.jpg"alt="CallCentreProbabilityGraph"style="width:500px;cursor:pointer"><h3>Example2:BakerySalesProbability</h3><p>Abakerytypicallysellsanaverageof16specialcakesperday.Calculatetheprobabilityofsellingmorethan20cakesonagivenday.</p><h4>Solution:</h4><ul><li><strong>Given:</strong> calls.</li></ul><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/8edee42e-1233-401c-bab5-38e832b7c69c-file.jpg" alt="Call Centre Probability Graph" style="width: 500px; cursor: pointer"><h3>Example 2: Bakery Sales Probability</h3><p>A bakery typically sells an average of 16 special cakes per day. Calculate the probability of selling more than 20 cakes on a given day.</p><h4>Solution:</h4><ul><li><strong>Given:</strong> \lambda = 16, X > 20.</li><li><strong>ApplyingContinuityCorrection:</strong>Adjustto. </li><li><strong>Applying Continuity Correction: </strong>Adjust to X = 20.5.</li><li><strong>Calculate:</strong>.</li><li><strong>Calculate: </strong> Z = \frac{20.5 - 16}{\sqrt{16}} = 1.125.</li><li><strong>Result:</strong>Probabilityfor.</li><li><strong>Result: </strong>Probability for Z \geq 1.125$ cakes is ~13.03%. So, 13.03% chance of selling ( > 20 ).
    Bakery Sales Probability
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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