Question

Individual customers arrive at a coffee shop after it opens at 6 AM. The number of customers arriving by 7AM has a Poisson distribution with parameter λ = 10. 25% of the customers buy a single donut and coffee, and 75% just buy coffee. Each customer acts independently. Answer the following:

a.What are the two random variables described above and what are their probability

mass functions?

b.What is the probability that exactly five customers arrive by 7AM?

c.If 10 customers enter by 7AM, what is the probability that five donuts are

purchased?

d.What is the expected number (true mean) of donuts a single customer will

purchase, and what is the expected number of customers that arrive by 10AM?

Answer 1

Part a.

The 2 random variables are,

X : the number of customers arriving by 7 AM.

Z : No. of donuts bought by X customers.

&

Part b.

Part c.

Let Z be the random variable denoting the number of donuts bought.

Then ,

To compute,

Part d.

expected number (true mean) of donuts a single customer will purchase is,

what is the expected number of customers that arrive by 10AM is,

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