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summary:Let two random vectors $X_1$ and $X_2$ be jointly distributed as a normal distribution with mean $\mu$ and covariance matrix $\sum_1$. Let $\Pi (\lambda)$be the probability that $X_1\in C_1, X_2\in C_2$, where $C_1$ and $C_2$ are convex symmetric sets, when the covariance matrix between $X_1$ and $X_2$ is multiplied by $\lambda;0\leq \lambda \leq 1$. It is shown that $\Pi(\lambda)$ increases with $\lambda$ under some conditions on $\mu$ and $\sum_1$. This generalizes the results of Das Gupta et al (1972), Khatri (1967) and Šidák (1973)

On Certain Inequalities for Normal Distributions and their Applications to Simultaneous Confidence Bounds