Step-by-step mathematical evaluation:
For a Poisson variate with parameter $\lambda$:

Given $P(X=1) = P(X=2)$:

Now, evaluate $P(X=4)$ with $\lambda = 2$:

• Incorrect: $(5e^{-2})/6$ is obtained from arithmetic error.
• Incorrect: $(2e^{-2})/2 = e^{-2}$, which corresponds to $P(X=2)$ or $P(X=1)$.
• Incorrect: $(e^{-2})/3$ misses a factor of 2.
• Correct: Evaluating $\frac{16}{24} e^{-2}$ gives exactly $\frac{2}{3}e^{-2}$.

Calculation of variance of binomial distribution:
Given $n = 10$, probability of failure $q = 0.3$.
Probability of success is $p = 1 - q = 1 - 0.3 = 0.7$.

• Incorrect: 21 misses the decimal point.
• Correct: The variance of a binomial distribution is given by:
  $$\text{Var}(X) = n \cdot p \cdot q = 10 \times 0.7 \times 0.3 = 2.1$$
• Incorrect: 7 is the mean of the distribution ($np = 10 \times 0.7 = 7$).
• Incorrect: 3 corresponds to $nq = 10 \times 0.3 = 3$.

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.