Statistical Methods & Experimental Designs

Calculation using binomial distribution:

• Given $n = 300$, probability of defective $p = 0.03$, and $q = 1 - p = 0.97$. The variance is $\sigma^2 = n p q = 300 \times 0.03 \times 0.97 = 9 \times 0.97 = 8.73$. The standard deviation is $\sigma = \sqrt{8.73} \approx 2.9546$.
• $8.73$ is the variance ($\sigma^2$), not the standard deviation.
• $9$ is the expected value / mean ($n p = 300 \times 0.03 = 9$).
• $3$ is a simple rounding of the root without precision.

Covariance in multinomial distribution:

• positive covariance is impossible because trials are constrained by $\sum X_i = n$; an increase in one outcome necessitates a decrease in others.
• In a multinomial distribution with $n$ trials and probabilities $p_1, \dots, p_k$, the marginal distribution of each $X_i$ is $\operatorname{Binomial}(n, p_i)$ with $\operatorname{Var}(X_i) = n p_i (1 - p_i)$. Because the sum $\sum_{i=1}^k X_i = n$ is fixed, the variables are negatively correlated, and the exact covariance between any two distinct categories $i \ne j$ is $\operatorname{Cov}(X_i, X_j) = -n p_i p_j$.
• Omits both the negative sign and sample size $n$.
• Lacks the sample size parameter $n$.

Mode of sum of independent binomial variates:

• $3, 2$ would imply bimodal distribution, which occurs only if $(n+1)p$ is an integer.
• By the additive property of the binomial distribution, since $p$ is identical, $Z = X + Y \sim \operatorname{Binomial}(n_1 + n_2, p) = \operatorname{Binomial}(4 + 5, 1/3) = \operatorname{Binomial}(9, 1/3)$. The mode is given by $\lfloor (n + 1)p \rfloor$. Here $(9 + 1) \times (1/3) = 10/3 = 3.333$. Since $10/3$ is non-integral, there is a unique mode equal to the integer part: $\lfloor 3.333 \rfloor = 3$.
• $3, 4$ is incorrect.
• $4$ is the ceiling, not the mode.