Question

Consider the following probability distributions.
(A) Normal distribution
(B) Binomial distribution
(C) Poisson distribution
(D) F-distribution
(E) Chi-square distribution
In which of the above distributions are mean and variance equal:

• (A) only.
• (B) only.
• (C) only.
• (D) and (E) only.

Hint:

Compare the standard mean and variance formula of each named distribution.

Answer 1

The Correct Option is C

Step 1: Recall the mean and variance for each distribution. For the Normal distribution (A), the mean is \(\mu\) and the variance is \(\sigma^2\), two independent parameters that are generally not equal.
Step 2: For the Binomial distribution (B) with parameters \(n\) and \(p\), mean \(= np\) and variance \(= np(1-p)\). Since \(0 < p < 1\), the factor \((1-p)\) is less than 1, so the variance is always strictly less than the mean.
Step 3: For the Poisson distribution (C) with parameter \(\lambda\), both the mean and the variance equal \(\lambda\). This equality is the defining property of the Poisson distribution and is used as a diagnostic check (equidispersion) for count data.
Step 4: For the F-distribution (D) the mean is \(\frac{d_2}{d_2-2}\) and the variance is a more complex expression in both degrees of freedom; these are never equal. For the Chi-square distribution (E) with \(k\) degrees of freedom, mean \(=k\) but variance \(=2k\), always twice the mean.
Step 5: Only the Poisson distribution (C) satisfies mean = variance, so the correct choice is (C) only, option 3.