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Variance is a measure that tells us how much a set of numbers is spread out. In simpler terms, it shows how much the values in a dataset differ from the average (or mean). In probability and statistics, understanding variance helps us understand how much a random variable deviates from its expected value. A higher variance indicates that the numbers are more spread out.
In the context of a Poisson distribution, variance has a special relationship. For Poisson distributions, the variance is equal to the parameter \( \lambda \), which also represents the mean. This means when \( \lambda \) is 2, both the mean and variance are 2.

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It helps us understand and predict the chances of events happening. In simple terms, it's a way to measure how likely something is to occur.
When applying probability theory to random variables like those found in Poisson distributions, it helps us understand how often we can expect certain results. Probability theory is fundamental in creating distributions, such as the Poisson, and is crucial in interpreting the chance or frequency of events in a given time period. Knowing the probability distributions, like Poisson, also assists in calculating probabilities of specific events.

A random variable is a key concept in statistics and probability theory. It is a variable that can take on different values, each associated with a certain probability. Random variables can be either discrete or continuous, with Poisson distribution dealing particularly with discrete random variables.
The outcome of a random variable is not fixed and can vary each time you observe or measure it. In the context of a Poisson distribution, a random variable represents the number of times an event occurs in a fixed interval of time or space, with events happening independently at a constant average rate. Understanding how random variables work help in predicting outcomes based on the probability distribution.

The mean of a distribution is essentially the average or expected value of a set of numbers. It gives a central value around which the numbers tend to cluster. For a Poisson distribution, the mean is determined by its parameter \( \lambda \).
In the case of a Poisson distribution where \( \lambda = 2 \), the mean is also 2. This dual role of \( \lambda \) in Poisson distributions makes it unique because it sets both the mean and variance of the distribution. The mean helps to tell us what value to expect on average and is a helpful guiding point for interpreting other statistical measures such as variance.