Explanation

For a Poisson distribution with mean λ:

1. Mean = λ
2. Variance = λ
3. Standard deviation = √λ
4. Median is approximately λ but not exactly equal to λ (it's approximately λ - 1/3 for large λ)

Therefore:

• Mean equals the variance (both are λ) - This is correct
• Mean does NOT equal the standard deviation (mean = λ, standard deviation = √λ)
• Median does NOT equal the variance
• Median, mean, and variance are NOT all equal

This is a key property of the Poisson distribution: the mean and variance are equal. This property distinguishes the Poisson distribution from other distributions like the normal distribution (where mean and variance are independent parameters) or the binomial distribution (where variance = np(1-p) ≠ mean = np).