Minimal property of standard deviation:

• Standard deviation is the minimum, never the maximum.
• The root mean square deviation about any arbitrary point $A$ is $S = \sqrt{\frac{1}{n} \sum (X_i - A)^2}$. By algebraic properties, $S^2 = \sigma^2 + (\bar{X} - A)^2$. Since $(\bar{X} - A)^2 \ge 0$, $S^2$ attains its unique absolute minimum when $A = \bar{X}$, meaning standard deviation is the least possible value of root mean square deviation.
• Standard deviation equals r.m.s.d. only when deviations are taken from the arithmetic mean, not universally.
• False proportional relationship.

Poisson probability mass function computation:

• $1$ would represent certainty, which is not true for $P(X=1)$.
• The Poisson probability formula is $P(X = k) = \frac{e^{-m} m^k}{k!}$. Substituting mean $m = 1$ and $k = 1$: $P(1) = \frac{e^{-1} (1)^1}{1!} = e^{-1} = \frac{1}{e}$.
• $e$ exceeds $1$, which violates the axiom that probability cannot exceed $1$.
• $e/2$ exceeds $1$, impossible for a probability.