Statistical Methods & Experimental Designs

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.

By Karl Pearson's moment inequality derived from the Cauchy-Schwarz inequality for any probability distribution:

where $\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ is the measure of skewness and $\beta_2 = \frac{\mu_4}{\mu_2^2}$ is the measure of kurtosis.
Since $\beta_1 \ge 0$ (being a ratio involving a squared real quantity), it strictly follows that:

When a single six-faced die is thrown, the number of possible outcomes is 6. When two independent unbiased dice are thrown simultaneously, the total number of outcomes in the sample space $S$ is given by the fundamental principle of counting:

The outcomes range from $(1,1), (1,2), \dots, (6,6)$.