Correlation of two sums sharing fifty terms

Question

Let $X_1,\dots,X_{150}$ be i.i.d. with variance $1$ . Form $U=3\sum_{i=1}^{50}X_i+\sum_{i=51}^{100}X_i,\qquad V=\sum_{i=51}^{100}X_i+3\sum_{i=101}^{150}X_i.$ What is the correlation between $U$ and $V$ ?

Short answer

$1/10$ . Only the shared block $X_{51},\dots,X_{100}$ contributes to the covariance, which is $50$ . Each sum has variance $9\cdot50+50=500$ , so the correlation is $50/500=0.1$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Covariance and Correlation
Populations, Samples, and Sampling Distributions

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