Random variables
A random variable X assigns a numerical value to each outcome of an experiment. A discrete random variable takes countable values such as 0, 1, 2, … (e.g. the number of heads in coin tosses). A continuous random variable can take any value in an interval (e.g. height).
Probability distribution
For a discrete X, the probability distribution lists P(X = x) for every value x. Two requirements must hold:
- 0 ≤ P(X = x) ≤ 1 for every x
- the probabilities add to 1
Key formula
Σ P(X = x) = 1. Use this to find an unknown probability or constant k.
Worked example
Worked example
A fair die is tossed and X is the score. Then P(X = x) = 1/6 for x = 1, 2, …, 6. So P(X > 4) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3.
If P(X = x) = kx for x = 1, 2, 3, 4, then Σ = k(1 + 2 + 3 + 4) = 10k = 1, giving k = 1/10.
Remember
- All probabilities must sum to exactly 1.
- Each probability lies between 0 and 1.
- P(X ≥ 1) = 1 − P(X = 0) is often quicker.