Cross-Correlation between Variables?

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If Pearson's correlation coefficient for variables $X_1, X_2$ is $p_1$, and the correlation coefficient for variables $X_2, X_3$ is $p_2$, can we use this information to reasonably estimate the correlation between $X_2$ and $X_3$?

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In case you mean the coefficient between $X_1$ and $X_3$ the answer is no as you don’t know the covariance between them.