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Chapter 5: Problem 58

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 \(X_{i}\) 's were not independent? Explain.

Short Answer

Expert verified
a. \(E(V) = 87850\) ft鲁, \(V(V) = 19050216\) ft鲁虏. b. The calculations depend on independence; variance would change with dependence.

Step by step solution

01

Determine the Expression for Total Volume

The total volume shipped is given as the sum of the volumes of each type of container multiplied by the number of each container. The expression is \(V = 27X_1 + 125X_2 + 512X_3\).
02

Calculate Expected Value of Total Volume

To find the expected value of the total volume, \(E(V)\), we use the linearity of expectation: \(E(V) = E(27X_1 + 125X_2 + 512X_3) = 27E(X_1) + 125E(X_2) + 512E(X_3)\). Substituting the given means, we have:\[E(V) = 27(200) + 125(250) + 512(100)\]. Calculate these to find the total expected volume.
03

Calculate Each Term for Expected Value

Calculate each term separately:- \(27 imes 200 = 5400\)- \(125 \times 250 = 31250\)- \(512 \times 100 = 51200\)Add these results to get the total expected value.
04

Sum the Expected Values

Add the results from Step 3 to get the total expected value of the volume:\[5400 + 31250 + 51200 = E(V) = 87850\] cubic feet.
05

Calculate Variance of Total Volume

Since the \(X_i\)'s are independent, the variance of the total volume \(V\) can be calculated as \(V(V) = V(27X_1 + 125X_2 + 512X_3) = 27^2V(X_1) + 125^2V(X_2) + 512^2V(X_3)\). Use \(V(X_i) = \sigma_i^2\) and calculate:- \(27^2 \times 10^2 = 72900\)- \(125^2 \times 12^2 = 2250000\)- \(512^2 \times 8^2 = 16777216\).Sum these values to obtain \(V(V)\).
06

Sum the Variance Terms

Add the variance terms calculated in Step 5 to find the total variance:\[72900 + 2250000 + 16777216 = V(V) = 19050216\] cubic feet squared.
07

Understand the Impact of Independence

The calculations for variance assume independence among the variables \(X_1, X_2, X_3\). If these variables are not independent, the covariance terms need to be included in the variance calculation, which would potentially alter the calculated variance. The expected value remains valid even without independence due to the linearity of expectation.

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Key Concepts

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

Expected Value
The expected value, often denoted as \(E(X)\), is a fundamental concept in probability and statistics. It is also known as the mean and represents the average outcome we would expect from a random variable over an infinite number of trials. In the context of shipping containers, the expected value helps us understand the average total volume of containers shipped each week.

To calculate the expected value of the total shipping volume, we consider the linearity of expectation, which states that the expected value of a sum is the sum of the expected values. This linear property simplifies calculations significantly.
  • The total expected volume is found using the formula: \(E(V) = 27E(X_1) + 125E(X_2) + 512E(X_3)\).
  • Using the given means where \(E(X_1) = 200\), \(E(X_2) = 250\), and \(E(X_3) = 100\), we compute each term.
For example, \(27 \times 200 = 5400\), \(125 \times 250 = 31250\), and \(512 \times 100 = 51200\). By adding these values, we get the total expected volume:
\(E(V) = 5400 + 31250 + 51200 = 87850\) cubic feet. So, the expected value tells us that on average, 87850 cubic feet of cargo is handled weekly.
Variance
Variance measures how much the values of a random variable differ from the expected value. It tells us about the spread or dispersion of the data around the mean. With the shipping volumes, variance helps us gauge the variability in the total volume shipped each week.

The formula for variance when dealing with independent random variables \(V = a_1X_1 + a_2X_2 + a_3X_3\) uses the variances of each individual variable without covariance terms, thanks to their independence. The variance formula is:
\(V(V) = a_1^2V(X_1) + a_2^2V(X_2) + a_3^2V(X_3)\)
  • For our problem, the variances are \(V(X_1) = \sigma_1^2 = 100\), \(V(X_2) = \sigma_2^2 = 144\), and \(V(X_3) = \sigma_3^2 = 64\).
  • The calculation would be \(27^2 \times 100 + 125^2 \times 144 + 512^2 \times 64\).
This calculation leads to individual results: \(72900\), \(2250000\), and \(16777216\). Summing these, we find the total variance:
\(V(V) = 19050216\) cubic feet squared. This significant variance indicates a substantial spread in the total volume shipped from week to week.
Independence in Random Variables
Independence in random variables is a condition where the outcome of one random variable doesn't affect the outcome of another. This concept is crucial for simplifying calculations in probability and statistics.

In our shipping company example, the independence of \(X_1\), \(X_2\), and \(X_3\) simplifies the calculation of the variance. If these variables were not independent, each pair would have a covariance term that would need to be included in the variance calculation.
  • Independence allows the variance to be the sum of the individual variances, without additional covariance computations.
However, even if the random variables are not independent, the expected value remains unchanged due to the linearity of expectation. Thus, independence plays a critical role in the variance calculation but is not necessary for computing the expected value. It's vital to understand this distinction as it affects the reliability of our calculations in real-world applications.

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Most popular questions from this chapter

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