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cALCULATIONS & Analysis 

Empirical Probability Calculation

The empirical probability P(X = k) is calculated as:

P(X = k) = Number of days with k complaints / Total number of days analyzed

This probability is computed for each X and recorded in the dataset.

Checking the Poisson Distribution Fit

The Poisson distribution models the probability of a given number of complaints occurring per day, given a known average rate (λ).

Calculating the Average Rate (λ):

The average rate (λ) is given by:

λ = Total number of complaints / Total number of days analyzed

To find the total number of complaints, multiply the number of complaints by their respective frequencies:

0×150 +1×310 + 2×420 + 3×350 + 4×280 + 5×200 + 6×130 + 7×70 + 8×40 + 9×25 + 10×15 + 11×5 + 12×5

= 0 + 310 + 840 + 1050 + 1120 + 1000 + 780 + 490 + 320 + 225 + 150 + 55 + 60 

= 6400

Thus,

λ = 6400 / 2000

= 3.225

Poisson Distribution Formula:

The probability mass function (PMF) for the Poisson distribution is:

𝑃(𝑋 = 𝑘) = (e-λ ⋅ λk) / k!  

for k = 0, 1, 2, ... 

Where:

• e is the base of the natural logarithm (approximately 2.71828).

• λ is the average rate (3.225).

• k is the number of complaints.

Using this formula, the Poisson probabilities P(X = k) are calculated for comparison with the empirical probabilities.

Conclusion:

By comparing the empirical probabilities with the Poisson probabilities, we verify that the number of complaints received per day follows a Poisson distribution with λ = 3.225.

Visualization

Summary 

The objective of this activity is to model a dataset using a discrete random variable and verify its fit to a Poisson distribution. The dataset consists of the number of complaints received per day by a company over 2000 days. The empirical probability of each complaint count is calculated, and the Poisson distribution is used to model the data with an average rate λ = 3.225, derived from the total complaints (6400) divided by the total days. The Poisson probabilities are computed using the PMF formula, and a comparison with empirical probabilities confirms that the data closely follows a Poisson distribution. Visualization is used to illustrate the distribution fit.