Final Answer:
To calculate the average (mean) of a discrete random variable, you can use the formula μ = Σ (x * P(x)), where x represents each possible value of the random variable and P(x) is the corresponding probability. The spread of data for a discrete random variable is measured using the variance (σ²) and standard deviation (σ), calculated as Var(X) = Σ [(x - μ)² * P(x)] and σ = √Var(X), respectively.
Step-by-step explanation:
The average (mean) of a discrete random variable is found by multiplying each possible value of the variable (x) by its corresponding probability (P(x)), summing up these products, and denoting the result as μ. Mathematically, this is expressed as μ = Σ (x * P(x)). For example, if you have a discrete random variable X with values 1, 2, and 3, and their respective probabilities are 0.2, 0.5, and 0.3, the mean would be calculated as μ = (1 * 0.2) + (2 * 0.5) + (3 * 0.3).
The spread of data for a discrete random variable is measured by variance (Var(X)) and standard deviation (σ), where Var(X) = Σ [(x - μ)² * P(x)]. Here, (x - μ) represents the deviation of each value from the mean, squared, and weighted by its probability. The standard deviation is then obtained by taking the square root of the variance, expressed as σ = √Var(X). These measures provide insights into how much the values of the random variable deviate from their average, helping to gauge the variability or dispersion of the data.
In summary, calculating the average involves determining the weighted sum of all possible values, while the spread of data is quantified through variance and standard deviation, considering the squared differences between individual values and the mean.