Final answer:
The question deals with the probability mass function of a discrete random variable, calculation of expected value, variance, and standard deviation of various expressions involving the random variable X.
Step-by-step explanation:
The student's question pertains to the probability mass function (p.m.f) of a discrete random variable X. The random variable X can take on values 0, 1, 2, 3, 4, 5. This probability distribution is discrete because the values are countable and the probabilities sum up to one, making it a valid probability distribution. The expected value E(X) or mean (μ) of X, is calculated by summing the products of each value of the random variable and its corresponding probability, represented by the formula E(X) = μ = Σ xP(x).
To find other expected values, such as E(3x² - 5) or E(x - 2), or to calculate the variance V(6x + 8) and the standard deviation σx, you would use the definitions and properties of expected value, variance, and standard deviation of a discrete random variable. For variance and standard deviation, the calculation involves finding the average of the squared deviations from the mean, adjusted by the linear transformation if present.