Statistical Methods & Experimental Designs

Analysis of mode in Poisson distribution:
In a Poisson distribution with parameter $\lambda$:

• Correct: When $\lambda$ is an integer (say $k$), the distribution is bimodal, and the two modes occur at values $x = k - 1$ and $x = k$, where $P(X = k - 1) = P(X = k)$.
• Incorrect: The distribution is unimodal only when $\lambda$ is not an integer, with the unique mode at $[\lambda]$ (integral part of $\lambda$).
• Incorrect: 'Positive modal' is not standard statistical terminology.
• Incorrect: 'Negative modal' is not standard statistical terminology.

Step-by-step binomial probability calculation:
Given $n = 3$, $p = 2/3$, so $q = 1 - p = 1/3$.
We need to compute $P(2 \le Y) = P(Y \ge 2) = P(Y=2) + P(Y=3)$.

• Correct:
  Using the binomial PMF $P(Y=k) = \binom{n}{k} p^k q^{n-k}$:
  $$P(Y=2) = \binom{3}{2} \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^1 = 3 \times \frac{4}{9} \times \frac{1}{3} = \frac{12}{27}$$
  $$P(Y=3) = \binom{3}{3} \left(\frac{2}{3}\right)^3 \left(\frac{1}{3}\right)^0 = 1 \times \frac{8}{27} \times 1 = \frac{8}{27}$$
  Summing these probabilities:
  $$P(Y \ge 2) = \frac{12}{27} + \frac{8}{27} = \frac{20}{27}$$
• Incorrect: $21/27$ is an arithmetic error.
• Incorrect: $22/27$ is an arithmetic error.
• Incorrect: $23/27$ is an arithmetic error.

Analysis of skewness in binomial distribution:
The coefficient of skewness (Karl Pearson's $\gamma_1$ or $\beta_1$) for a binomial distribution $B(n, p)$ is:

• Incorrect: When $p \lt 1/2$, $q \gt p$, so $\gamma_1 \gt 0$ (positively skewed).
• Correct: When $p = 1/2$, $q = 1/2$, giving $q - p = 0$ and $\gamma_1 = 0$; hence the distribution is perfectly symmetrical and skewness is zero.
• Incorrect: When $p \gt 1/2$, $q \lt p$, so $\gamma_1 \lt 0$ (negatively skewed).
• Incorrect: $p \lt q$ is identical to $p \lt 1/2$, giving positive skewness.

Step-by-step variance evaluation:
For any Poisson variate, the variance equals the mean:

• Incorrect: 6 is obtained by calculating $3 \times 1 + 3 = 6$, which is the mean $E(3X+Y)$, not the variance.
• Incorrect: 9 is $3^2 \text{Var}(X)$.
• Correct: For independent random variables $X$ and $Y$:
  $$\text{Var}(3X + Y) = 3^2 \text{Var}(X) + 1^2 \text{Var}(Y) = 9(1) + 1(3) = 9 + 3 = 12$$
• Incorrect: 18 is obtained if the coefficients are squared incorrectly.