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A shipping company handles containers in three different sizes: (1) \(27 \mathrm{ft}^{3}(3 \times 3 \times 3)\), (2) \(125 \mathrm{ft}^{3}\), and (3) \(512 \mathrm{ft}^{3}\). Let \(X_{i}(i=1,2,3)\) denote the number of type \(i\) containers shipped during a given week. With \(\mu_{i}=E\left(X_{i}\right)\) and \(\sigma_{i}^{2}=V\left(X_{i}\right)\), suppose that the mean values and standard deviations are as follows: $$ \begin{array}{lll} \mu_{1}=200 & \mu_{2}=250 & \mu_{3}=100 \\ \sigma_{1}=10 & \sigma_{2}=12 & \sigma_{3}=8 \end{array} $$ a. Assuming that \(X_{1}, X_{2}, X_{3}\) are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume \(=27 X_{1}+125 X_{2}+512 X_{3}\).] b. Would your calculations necessarily be correct if the \(X_{i}^{\prime}\) s were not independent? Explain.

Step by step solution

01

Understand the Scenario

The problem involves calculating the expected total volume shipped and its variance for different container types. The sizes and shipping distributions of the containers are given. The task is to use these data to find the expected total volume and variance of the shipments.

02

Set Up the Total Volume Formula

The total volume shipped is calculated by combining the volumes of each container type, using the formula: \(V = 27X_1 + 125X_2 + 512X_3\).

03

Calculate the Expected Total Volume

The expected value of the total volume is calculated using the linearity of expectation: \[E(V) = E(27X_1 + 125X_2 + 512X_3) = 27E(X_1) + 125E(X_2) + 512E(X_3) = 27\mu_1 + 125\mu_2 + 512\mu_3.\]Substituting the values, we get: \(E(V) = 27(200) + 125(250) + 512(100)\).

04

Compute the Expected Total Volume

Calculate the actual expected value:\(E(V) = 5400 + 31250 + 51200 = 87850\ \text{ft}^{3}.\)

05

Calculate the Variance of the Total Volume

Assuming independence between \(X_1, X_2, X_3\), the variance of the total volume is:\[V(V) = V(27X_1 + 125X_2 + 512X_3) = 27^2V(X_1) + 125^2V(X_2) + 512^2V(X_3).\]Since \(V(X_i) = \sigma_i^2\), this becomes:\(= 27^2\sigma_1^2 + 125^2\sigma_2^2 + 512^2\sigma_3^2\).

06

Compute the Variance of the Total Volume

Calculate each variance component:\( 27^2\sigma_1^2 = 729 \times 10^2 = 72900, \)\( 125^2\sigma_2^2 = 15625 \times 12^2 = 2250000, \)\( 512^2\sigma_3^2 = 262144 \times 8^2 = 16777216. \)Summing these gives:\(V(V) = 72900 + 2250000 + 16777216 = 19040116\ \text{ft}^6.\)

07

Consider Dependency of Variables

If the \(X_i\) variables were not independent, the variance calculation would have additional terms relating to the covariance between these variables. Thus, the variance result is valid only under the assumption of independence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Container Volume Calculation

When dealing with container volume calculations, it's essential to consider both the distinct dimensions and how each type of container contributes to the overall volume. For instance, the given dimensions are quite specific: a cube-shaped container, a moderately larger one, and a significantly larger container. These are:

• A cube with dimensions of 3 feet per side, giving a volume of \(27 \mathrm{ft}^3\).
• A container of \(125 \mathrm{ft}^3\), presumably not cube-shaped.
• A much larger container at \(512 \mathrm{ft}^3\).

Each type of container can be shipped in differing quantities, represented by random variables \(X_i\). The formula for total volume combines these quantities by multiplying the volume of each type by its shipped amount and adding the results:
\[V = 27X_1 + 125X_2 + 512X_3\].
This equation allows us to compute the total volume managed by the shipping company within a specific timeframe.

Independence Assumption in Probability

Independence is a fundamental assumption in probability that states the occurrence of one event does not affect the occurrence of another. In our exercise, we assume independence among the number of containers shipped for each type, \(X_1, X_2, X_3\).
This independence matters because:

• It ensures our calculation of variance is straightforward, as there are no covariances among the variables.
• If the variables were dependent, covariances would introduce complexity, affecting the variance and total calculated values.
• Independence simplifies the use of basic probability rules, ensuring reliable computation of total volume expectations and variances.

Consequently, in practical applications, verifying the independence of random variables can validate or invalidate statistical assumptions such as our variance calculations.

Linearity of Expectation in Statistics

Linearity of expectation is a valuable principle in statistics, allowing us to find expectations easily鈥攅ven for complex sums of random variables. This principle asserts that the expectation of a sum is the sum of expectations, regardless of any independence among the variables.
In our scenario, we apply linearity to compute the expected value of the total container volume:
\[E(V) = 27E(X_1) + 125E(X_2) + 512E(X_3)\].
This equation benefits from the property, as each component can be independently calculated and then summed to find the overall expected total volume, \(87850 \mathrm{ft}^3\) in this instance.

• This principle holds true under all conditions, making it a robust tool for expectations calculations.
• Understanding linearity of expectation helps in efficiently breaking down and solving broader statistical problems.

By breaking down complex problems into simpler calculations, we gain clear insights into expected values that would otherwise be more challenging.