Statistics Notes on Probability Distributions and Models

Zucchini Length Probability Distribution

  • Harvesting Details:
    • Zucchinis are harvested between lengths of 190 mm to 270 mm.
    • Most common harvested length is 220 mm.
(i) Probability of a Zucchini being < 220 mm

  • Parameters:

    • Lower bound (a) = 190 mm
    • Upper bound (b) = 270 mm
    • Length at which probability is calculated (x) = 220 mm

  • Uniform Distribution Formula:
    f(x)=2b−af(x) = \frac{2}{b-a}

  • Calculating:

    • f(x)=2270−190=280=0.025f(x) = \frac{2}{270 - 190} = \frac{2}{80} = 0.025

  • Probability Estimate:

    • P(X < 220) = 30 imes 0.025 = 0.375
(ii) Median Length Comparison
  • Median Explanation:
    • The median is the middle value where 50% of probability lies below it.
  • Finding Median:
    • Since 0.375ext(or37.5%)0.375 ext{ (or 37.5\%)} of zucchinis are below 220 mm, the median length must be greater than 220 mm.
  • Conclusion:
    • The median length is more than 220 mm since only 37.5% are below.

ACL Surgery Time Probability Distribution

  • Surgery Details:
    • Typical duration of ACL surgery is between 120 and 150 minutes.
    • Most common surgery time is 130 minutes.
(i) Probability of Surgery Duration < 130 minutes
  • Parameters:
    • Lower bound (a) = 120 minutes
    • Upper bound (b) = 150 minutes
    • Time (x) = 130 minutes
  • Calculating:
    • f(x)=2b−a=2150−120=230=0.0667f(x) = \frac{2}{b-a} = \frac{2}{150-120} = \frac{2}{30} = 0.0667
    • P(X < 130) = 10 \times 0.0667 = 0.3335
(ii) Probability of Surgery Duration > 130 minutes given < 140 minutes
  • Relevant Values:
    • Duration < 140 minutes, Length 130 to 140 minutes
  • Calculating:
    • P(130 < X < 140) = 0.0667 + 0.0333 = 0.1
  • Finding P(X > 130 | X < 140):
    • P(X > 130 | X < 140) =\frac{P(130 < X < 140)}{P(X < 140)} = \frac{0.1}{0.8335} \approx 0.5999

Wilding Pines Re-emergence Probability

  • Parameters:
    • North Island: Return time between 3 to 18 months.
    • South Island: Return time between 3 to 27 months, average return at 12 months.
(i) Probability of Both Islands for Re-emergence < 8 months
  • Calculating North Island:
    • P(3 < x < 8) = \frac{5}{15} = 0.3333
  • Calculating South Island:
    • P(3 < x < 8) = \frac{2}{27} \approx 0.0741
  • Combined Probability:
    • P(Both)=P(North)×P(South)=0.3333×0.0741≈0.0247P(Both) = P(North) \times P(South) = 0.3333 \times 0.0741 \approx 0.0247
(ii) Independence of Probabilities
  • Assumption:
    • The model assumes independence between the North and South Island re-emergences.
  • Validity Check:
    • This assumption is likely inaccurate since weather and other environmental factors may simultaneously influence both regions, suggesting possible correlation.

Application of Poisson Model for Wilding Pines

  • Area of Rangitoto:
    • Total area = 2311 hectares (or 23.11 km²).
  • Reasons Against Poisson Model:
    • (1) Poisson assumes independence in occurrence events; however, re-establishment of wilding pines could be influenced by similar ecological conditions across the island.
    • (2) Poisson distribution is best suited for rare events; pines may have established a population that is not rare, violating the 'rare' event criteria.

Sketching Probability Distribution Models

  • North Island Probability Distribution:
    • Rectangular model reflecting the distribution of wilding pine return times.
  • South Island Probability Distribution:
    • Triangular model demonstrating variations in return time.