Quick questions on Discrete random variables explained: H2 Mathematics Probability and Statistics

A discrete random variable $X$ takes a countable set of values, each with a probability $\mathrm{P}(X = x)$. A valid distribution satisfies:

The expectation (mean) is the long-run average value:

Many H2 questions describe a situation and ask you to build the distribution table before computing anything. The routine is: list every value the variable can take, find the probability of each from the scenario (using counting or basic probability), tabulate them, and check the probabilities sum to $1$. For the number of heads in two coin tosses, the values are $0, 1, 2$ with probabilities $\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}$. Constructing the table correctly is the foundation everything else rests on, because a single wrong probability throws off both the expectation and the variance that follow.

A variable has $\mathrm{P}(X = 0) = 0.5$, $\mathrm{P}(X = 1) = 0.5$. Find $\mathrm{E}(X)$. [1 mark]

Given $\mathrm{E}(X) = 4$, find $\mathrm{E}(2X + 3)$. [2 marks]

Given $\operatorname{Var}(X) = 5$, find $\operatorname{Var}(2X - 1)$. [2 marks]