Statistical Methods & Experimental Designs

This follows a Binomial distribution $B(n = 10, p = 0.16)$:
Probability of failure $q = 1 - p = 1 - 0.16 = 0.84$
$\text{Mean} = np = 10 \times 0.16 = 1.6$
$\text{Variance} = npq = 10 \times 0.16 \times 0.84 = 1.344$
$\text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} = \sqrt{1.344} \approx 1.1593$

For a Poisson distribution with parameter $\lambda$, the probability mass function is $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$.

Given $P(X = 1) = P(X = 3)$:

The mode of a Poisson distribution when $\lambda$ is non-integral is the integer part of $\lambda$:

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 $\frac{2}{3}e^{-2}$.