Determine and use the variance of a discrete random variable as a measure of spread, 𝑉𝑎𝑟 (𝑋) = ∑ 𝑝𝑖 (𝑥𝑖 − 𝜇)2 where 𝑝𝑖 is the probability of outcome 𝑥𝑖 occurring, 𝜇 is the mean.

A discrete random variable X has a probability distribution with mean μ = 3.2 and E(X²) = 11.8. Explain what the variance of X measures in the context of this distribution, and determine its value.

Consider the Bernoulli distribution where the outcomes for rolling a standard die are obtaining a five and not obtaining a five. Determine the variance of the resulting Bernoulli distribution in this scenario.

A discrete random variable $X$ represents the number of faulty items in a sample of 5 products from a manufacturing line. The probability distribution is given in the table below. Determine the variance of $X$, correct to 2 decimal places.

A discrete random variable $X$ represents the number of defective items in a batch of 5 items. The probability distribution is given in the table below. Determine the variance of $X$, giving your answer correct to 2 decimal places.