Let X = the number of coffee cups that a waiter drops in a week and assume X ~ Poi(0.59).



Let T be the total cost to the company from coffee cups dropped by the waiter in a week and let T=2X since the coffee cups cost $2 each.



Calculate Var(T) correct to 2dp.



Given:

- $ X \sim \text{Poisson}(\lambda = 0.59) $, where $ X $ represents the number of coffee cups dropped in a week.
- The cost $ T = 2X $, where each dropped cup costs $2.

To find the variance of $ T $, we can use the properties of the Poisson distribution and linear transformations.

### Step 1: Variance of $ X $
For a Poisson random variable $ X $ with parameter $ \lambda $, the variance is given by:

\[
\text{Var}(X) = \lambda = 0.59.
\]

### Step 2: Variance of $ T = 2X $
If $ T = aX $, where $ a $ is a constant, the variance of $ T $ is given by:

\[
\text{Var}(T) = a^2 \cdot \text{Var}(X).
\]

Substituting $ a = 2 $ and $ \text{Var}(X) = 0.59 $:

\[
\text{Var}(T) = 2^2 \cdot 0.59 = 4 \cdot 0.59 = 2.36.
\]

### Final Answer
\[
\text{Var}(T) = 2.36.
\]

Would you like to explore more details or have any questions? Here are some related questions:

1. What is the expected value of $ T $?
2. How does the distribution of $ T $ look like?
3. How would the variance change if the cost per cup changed?
4. What is the probability that the waiter drops more than 1 cup in a week?
5. How do changes in the Poisson rate parameter $ \lambda $ affect the variance of $ T $?

**Tip:** For Poisson distributions, the mean and variance are always equal, simplifying many calculations!

Let X = the number of coffee cups that a waiter drops in a week and assume X ~ Poi(0.59). Let T be the total cost to the company from coffee cups dropped by the waiter in a week and let T=2X since the coffee cups cost $2 each. Calculate Var(T) correct to 2dp.

Learn how to calculate the variance for a Poisson-distributed variable X ~ Poi(0.59) and its linear transformation T = 2X, where each dropped coffee cup costs $2. The solution explains key concepts such as variance of Poisson distribution and linear transformations. Suitable for undergraduate statistics students.

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