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) 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,X≤20.
- Applying Continuity Correction: Adjust X to 20.5.
- Converting to Z-Score:
- $Z = \frac{X - λ}{√λ}
</li></ul></li><li><strong>Calculate:</strong>Z = \frac{20.5 - 18}{\sqrt{18}} \approx 0.59.</li><li><strong>Result:</strong>ProbabilityforZ = 0.59is 72.24\leq 20calls.</li></ul><imgsrc="https://tutorchase−production.s3.eu−west−2.amazonaws.com/8edee42e−1233−401c−bab5−38e832b7c69c−file.jpg"alt="CallCentreProbabilityGraph"style="width:500px;height:320px"width="500"height="320"><h3>Example2:BakerySalesProbability</h3><p>Abakerytypicallysellsanaverageof16specialcakesperday.Calculatetheprobabilityofsellingmorethan20cakesonagivenday.</p><h4>Solution:</h4><ul><li><strong>Given:</strong>\lambda = 16, X > 20.</li><li><strong>ApplyingContinuityCorrection:</strong>AdjusttoX = 20.5.</li><li><strong>Calculate:</strong>Z = \frac{20.5 - 16}{\sqrt{16}} = 1.125.</li><li><strong>Result:</strong>ProbabilityforZ \geq 1.125$ cakes is ~13.03%. So, 13.03% chance of selling ( > 20 ).
