The number of ice creams sold per hour from Mr Fishy’s van is observed to be a Poisson random variable with parameter 𝜆=8 Each ice cream costs two pounds but Mr Fishy has to pay five pounds per hour for the pitch. What is the variance of his hourly profit (measured in pounds and ignoring all other income and expenses)?

The profit per hour from Mr. Fishy's ice cream van is given by the formula: Profit = (Number of ice creams sold) * (Profit per ice cream) - (Pitch cost) Given that the number of ice creams sold per hour follows a Poisson distribution with parameter 𝜆 = 8, and the profit per ice cream is 2 pounds, and the pitch cost is 5 pounds, we can calculate the variance of his hourly profit. The Poisson distribution has a variance equal to its parameter 𝜆. So, the variance of the number of ice creams sold is: Variance of ice creams sold = 𝜆 = 8 Now, let's calculate the profit per hour: Profit = (Number of ice creams sold) * (Profit per ice cream) - (Pitch cost) Profit = X * 2 - 5 Where X is the number of ice creams sold. Since X follows a Poisson distribution with parameter 𝜆 = 8, we can use the properties of variance to find the variance of the profit: Variance of Profit = Variance of (X * 2 - 5) Variance of Profit = Variance of (2X) (since the variance of a constant is 0) Variance of Profit = 2^2 * Variance of X (since the variance of a constant times a random variable is the constant squared times the variance of the random variable) Substituting the Variance of X (which is 𝜆 = 8): Variance of Profit = 4 * 8 Variance of Profit = 32 So, the variance of Mr. Fishy's hourly profit is 32 pounds.