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The Poisson distribution is a probability model that helps us understand how often a particular event happens within a fixed interval of time or space. In the exercise, it is used to model the number of customers entering a store on a given day. This type of distribution is characterized by a parameter \( \lambda \), which represents the average rate of occurrence over that interval.

For instance, if the average number of customers entering the store (our \( \lambda \)) is 10, this means that we can expect about 10 customers on any given day. The Poisson distribution is very useful in situations where events occur independently. The formula for the Poisson probability of observing \( k \) events in an interval is:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
where \( k \) is the actual number of occurrences. By applying this formula, you can predict the likelihood of various numbers of customers visiting the store.

Uniform distribution is a type of probability distribution where every outcome between a lower and upper limit is equally likely. In this exercise, the amount of money a customer spends is described by a uniform distribution between 0 and 100. This means every dollar amount from 0 to 100 is just as likely as any other.

This type of distribution is characterized by two parameters, \( a \) and \( b \), which are the minimum and maximum values respectively. The probability of any single value in a continuous uniform distribution is technically zero, but the probability of a value falling within a specific range can be found. A practical example: If a customer spends uniformly between 0 to 100, the expected value would be calculated simply as the midpoint, \( 50 \).

The uniform distribution is quite straightforward and is particularly useful in situations where you have no reason to prefer one outcome over any other.

Expected value is a key concept in probability, representing the average or mean value a random variable takes in the long run. It provides a measure of the center of a distribution.

In probability models, calculating the expected value of a random event helps predict the long-term outcome of random processes. For a uniform distribution, like the spending of a customer, it can be found by averaging the minimum and maximum amounts.

The expected value \( E[X] \) is given by the formula: \[ E[X] = \frac{a + b}{2} \]
For our exercise, with a customer spending between 0 to 100 dollars, the expected amount spent is:
\[ E[X] = \frac{0 + 100}{2} = 50 \]
Thus, this informs us that on average, a customer will spend \(50. In the store's context, if 10 customers visited daily, the overall expected income would be 10 times that amount, equaling \)500.

Variance is a valuable statistical measure that can indicate how much outcomes deviate from the expected value, essentially representing the spread of a distribution. A small variance means data points are close to the mean, whereas a large variance indicates widespread values.

For a uniform distribution, variance can be calculated using the formula: \[ Var(X) = \frac{(b - a)^2}{12} \]
In the context of the exercise, customers spend uniformly between 0 to 100 dollars, so the variance is:\[ Var(X) = \frac{(100 - 0)^2}{12} = 833.33 \]
This tells us there is some variability in the amount spent by each customer. Understanding variance helps businesses anticipate fluctuations.

When evaluating the store's revenue, this variance, combined with the Poisson distribution of the customer flow, provides insights into potential variability in daily income, further described by the formula used for the store's overall variance.