Question

In which of the following distribution, mean is equal to variance ?

• A. t distribution
• B. Poisson distribution
• C. Binomial distribution
• D. Chi-square distribution

Hint:

A key defining property of the Poisson distribution is its equidispersion, meaning the mean is equal to the variance (\(\mu = \sigma^2 = \lambda\)). This property is often used to test if data follow a Poisson distribution.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Probability distributions are mathematical functions that describe the likelihood of different outcomes.
Each distribution is characterized by specific parameters, such as its mean (measure of central tendency) and variance (measure of dispersion).

Step 2: Detailed Explanation:
Let us analyze the relationship between mean (\(\mu\)) and variance (\(\sigma^2\)) for the given distributions:
- t-distribution: A symmetric, bell-shaped distribution. Its mean is 0, and its variance is given by \( \frac{v}{v - 2} \) (where \( v \) represents the degrees of freedom, for \( v > 2 \)). Thus, mean does not equal variance.
- Poisson distribution: A discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space.
The probability mass function is defined by a single parameter, lambda (\(\lambda\)), which represents the average rate of occurrence.
In a Poisson distribution, both the mathematical expectation (mean) and the variance are equal to this parameter:
\[ \text{Mean} (\mu) = \lambda \]
\[ \text{Variance} (\sigma^2) = \lambda \]
Therefore, mean is exactly equal to variance.
- Binomial distribution: A discrete distribution characterized by two parameters: the number of trials (\(n\)) and the probability of success (\(p\)). Its mean is \(np\), and its variance is \(np(1-p)\). Since \(1-p < 1\), the variance is always less than the mean.
- Chi-square distribution: A continuous distribution characterized by its degrees of freedom (\(k\)). Its mean is \(k\), and its variance is \(2k\).
Therefore, the Poisson distribution is the only distribution where the mean is equal to the variance.

Step 4: Final Answer:
The probability distribution in which the mean is equal to the variance is the Poisson distribution.