Mean Value as the Sole Parameter of the Distribution

The correct answer is option 'D': Poisson distribution. Let's explore why the mean value is the sole parameter of the Poisson distribution.

1. Introduction to Probability Distributions
Probability distributions are mathematical functions that describe the likelihood of different outcomes in an event or experiment. They are used to model and analyze various phenomena and are an essential tool in statistics.

2. The Mean Value
The mean value of a probability distribution represents the average or expected value of the random variable being studied. It provides a measure of central tendency and is often denoted by the Greek letter µ (mu).

3. Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. It is characterized by two parameters: the mean (µ) and the standard deviation (σ). The mean value is a parameter of the normal distribution, but it is not the sole parameter since the standard deviation also plays a crucial role.

4. Binomial Distribution
The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). The mean value of a binomial distribution can be calculated as the product of the number of trials and the probability of success.

5. Exponential Distribution
The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is characterized by a single parameter: the rate parameter (λ). The mean value of an exponential distribution is equal to the reciprocal of the rate parameter, i.e., 1/λ. Therefore, the mean value alone does not uniquely determine the distribution.

6. Poisson Distribution
The Poisson distribution models the number of events that occur in a fixed interval of time or space when these events occur with a known average rate and independently of the time since the last event. It is characterized by a single parameter: the rate parameter (λ), which represents the average number of events in the given interval. The mean value of a Poisson distribution is equal to the rate parameter (µ = λ). Therefore, the mean value is the sole parameter of the Poisson distribution.

Conclusion
Among the given probability distributions, the Poisson distribution is the only one where the mean value is the sole parameter of the distribution. In other distributions like the normal, binomial, and exponential distributions, additional parameters such as the standard deviation, number of trials, or rate parameter are required to fully describe the distribution.