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Variance is a crucial concept in statistics that measures how much a set of values spread out from their mean. It gives us insight into the variability or diversity of the data set.
Variance is symbolized by \( \sigma^2 \) and calculated using the formula:

• For a population: \( \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \)
• For a sample: \( s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \)

This formula captures how each data point \(x_i\) deviates from the mean, \(\mu\) (or \(\bar{x}\) for a sample).
The more spread out the data, the larger the variance. A variance of zero indicates that all data values are identical. It's essential to remember that variance is always non-negative.
For random variables, when considering the sum of variables like in our exercise, the variance of their total depends heavily on their independence.

The mean is a measure of central tendency, also known as the average. It provides a central point around which the numbers tend to cluster. Calculated simply by summing all values and dividing by the count, the mean is symbolized as \( \mu \) for a population and \( \bar{x} \) for a sample.

• Population mean: \( \mu = \frac{\sum_{i=1}^{N} x_i}{N} \)
• Sample mean: \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \)

When dealing with independent random variables, the mean of the sum is just the sum of their means.
This is vital for exercises like ours, where we want to evaluate the mean rainfall combining sources.

Independent random variables are variables whose outcomes do not influence each other. This means that knowing the outcome of one variable gives you no information about the other. This property significantly simplifies the statistical calculations we might perform.For two independent variables, \( X \) and \( Y \), these principles hold:

• The expected value (mean) of their sum is the sum of their expected values: \( E(X + Y) = E(X) + E(Y) \).
• The variance of their sum is the sum of their variances: \( Var(X + Y) = Var(X) + Var(Y) \).

This independence is central in our exercise, determining both the mean and variance of total rainfall from two different sources.

Probability theory forms the basis for understanding how likely events are to occur. It helps us model and predict various phenomena in the real world through random variables and probabilistic approaches. Random variables can be thought of as functions that assign a numerical value to each outcome in a sample space. Key concepts in probability theory include:

• Probability Mass Function (PMF) for discrete variables, which gives the probability of each outcome.
• Probability Density Function (PDF) for continuous variables, which describes the likelihood of different outcomes within a continuous range.
• Cumulative Distribution Function (CDF), which gives the probability that a random variable is less than or equal to a certain value.

Understanding these functions enables us to work with means and variances of random variables as demonstrated in probability exercises, ensuring that we can effectively calculate combined measures like in our given scenario.