**Question: **The number of customers entering a store on a given day is Poisson distributed with mean λ = 10. The amount of money spent by a customer is uniformly distributed over (0, 100). Find the mean and variance of the amount of money that the store takes in on a given day.

**Analytical Solution**

Calculation-wise, this is straightforward

- Expectation: As seen from the question, the variables are independent and hence, the expectation of the product is the product of the expectations which is λ*50 = 1*50 = 500
- Variance: This can be derived from the following formula. The variance is simply the sum of the following two quantities
- Expected Number of Customers Per Day * Variance of the amount spent per day by a customer
- Square of the expectation of the Number of Customers Per day * Variance of the amount spent per day by the customer
- This means the variance is $33,333

- Details on the calculation from step 2

**Simulation Solution**

With a few lines of the code, we can simulate this scenario, I have shown the same simulation with 5 kinds of “ChartElementFunction” to give a general idea about the quantiles. *Notice the asymmetry in the distributions. Can you guess why? (Psst. It’s because of one of the distributions in the problem I will let you figure out which one)*

**Code**

I used slot operators to condense the code. Take your time to break it down further if needed.

Module[
{
customers := RandomVariate[PoissonDistribution[10]],
spending := RandomVariate[UniformDistribution[{0, 100}]],
simulation,
mean,
variance
},
simulation =
Table[
Plus @@ (spending & /@ Range[customers]), 1000, 100];
mean = Mean /@ simulation;
variance = Variance /@ simulation;
Column[
Table[
Labeled[
Framed[
Row[
MapThread[
DistributionChart[#1,
ChartElementFunction -> chartElementFunction,
ImageSize -> 394, AspectRatio -> 0.35, PlotLabel -> #2,
PlotRange -> All] &,
{{mean, variance}, {"Mean:\[Mu]", "Variance:\!\(\*SuperscriptBox[\(\[Sigma]\), \(2\)]\)"}}]]],
Rotate[chartElementFunction, Divide[Pi, 2]], Left],
{chartElementFunction, {"PointDensity", "SmoothDensity", "GlassQuantile", "BoxWhisker", "HistogramDensity"}}
]
]
]

End of the post 🙂

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