9709 · 5.4
Discrete random variables
A discrete random variable is just a number that changes by chance, like the score on a die. We can map out its probabilities and calculate its average outcome (expectation) and how spread out the results are (variance).
Need to know
What you need to know
- For any valid probability distribution, two conditions must be met:
- 1. The probability of each outcome must be between 0 and 1 inclusive: $0 \le P(X=x) \le 1$.
- 2. The sum of all probabilities must be exactly 1: $\sum P(X=x) = 1$.
Explanation
The Blueprint of Chance
- A discrete variable X has specific values 'x'. Each value has a probability P(X=x), and all probabilities must sum to 1.
- The expectation E(X) is the weighted average. Multiply each value 'x' by its probability and sum them up: E(X) = Σx P(X=x).
- The variance Var(X) measures spread. First find E(X²), the average of the squared values, then use Var(X) = E(X²) − [E(X)]².
- If you transform X into a new variable Y = aX + b, the new expectation is E(Y) = aE(X) + b, but the new variance is Var(Y) = a²Var(X).