Mean and Variance of Discrete Distributions (AS)

Discrete random variables are those that can take on only specific values, often integers, within a defined range. An example of a discrete random variable could be the number of heads tossed in 3 coin tosses.
The mean of a discrete random variable, typically denoted by µ (mu), is defined as the sum of all the possible values the variable can take, each multiplied by their respective probability. The calculation of the mean, or expected value, involves the formula: µ = Σ[x * P(X=x)].
The variance, denoted by σ^2 (sigma-squared), measures the dispersion or variability of the data points from the mean. It is calculated as: σ^2 = Σ[(x - µ)^2 * P(X=x)]. This formula indicates that we determine the square of each value’s deviation from the mean, then multiply each by the corresponding probability and sum the results.
The standard deviation is the square root of the variance and shows the average difference between each data value and the mean.
Remember that for discrete probability distributions, the sum of all probabilities must equal 1.
If you add or subtract a constant to every outcome of a variable, the mean will change by that amount, but the variance will remain the same. For example, if you add 3 to every outcome for a random variable X, the mean of the new distribution will be µ + 3, but the variance will be unchanged.
If you multiply every outcome of a variable by a constant, both the mean and the variance will be multiplied by the square of that constant.
Be familiar with specific discrete probability distributions, including the Uniform distribution, Bernoulli distribution, and Binomial distribution. Each has specific formulas to calculate the mean and variance, reflecting the different probabilities assigned to the possible outcomes.
Questions involving the mean and variance of discrete distributions often require setting up equations using these definitions and solving for unknowns. Practice is key to becoming comfortable with this process.
Graphs, tables, or probability mass functions might be used to present distribution data. Be prepared to interpret these and to use them in your calculations.