Question:
<p>If a variate <em>X</em> is expressed as a linear function of two variates <em>U</em> and <em>V</em> in the form \(X = aU + bV\), then the mean \(\bar{X}\) of <em>X</em> is</p>
<p>\(a\bar{U} - b\bar{V}\)</p>
<p>\(\bar{U} + \bar{V}\)</p>
<p>\(b\bar{U} + a\bar{V}\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: The expectation operator (mean) is linear, so E(aU + bV) = aE(U) + bE(V). This means the mean of a linear combination equals the same linear combination of the individual means.
<p><strong>Step 1:</strong> Recall that the mean (expectation) is a linear operator.</p><p><strong>Step 2:</strong> For any linear combination X = aU + bV, we have:</p><p>E(X) = E(aU + bV)</p><p><strong>Step 3:</strong> Apply linearity of expectation:</p><p>E(aU + bV) = aE(U) + bE(V)</p><p><strong>Step 4:</strong> Substitute the means:</p><p>E(X) = a·̄U + b·̄V</p><p>∴ Answer: <strong>D</strong> (The mean of X is āU + b̄V)</p>
Correct Answer: D