Probability is the area under the curve: For a continuous variable (like a waiting time or a length), single values have probability 0 — instead probability lives in areas under a curve f(x).
Two rules make f(x) a genuine probability density function
- it is never negative (an area below the axis would be a negative probability), and
- the total area under it is exactly 1 (something is certain to happen).
Probability of an interval: The probability that X lands in an interval is just the area of that strip.
IB-style question — probability as an area
A continuous random variable X has pdf f(x) = (3/8)x² for 0 ≤ x ≤ 2, and f(x) = 0 elsewhere.
Find P(0 ≤ X ≤ 1).
Step by step
- The probability is the area, so integrate f from 0 to 1.
- Integrate: the antiderivative of x² is x³/3, so (3/8)(x³/3) = x³/8.
- Evaluate at the limits.
Final answer
P(0 ≤ X ≤ 1) = 1/8 = 0.125.
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Integrate over the whole support, set it equal to 1: Many exam pdfs come with an unknown constant, e.g. f(x) = kx² on some interval. Here is how to find k.
Finding k
Integrate
Integrate f over the interval where it's non-zero.
Set equal to 1
The total area must be 1, so set the answer equal to 1. This always gives one equation in one unknown.
Solve
Solve that equation for k.
Integrate, set = 1, solve.
Exam pattern: This is the single most common opening part of a continuous-distribution question.
IB-style question — find k
A continuous random variable X has pdf f(x) = kx for 0 ≤ x ≤ 4, and f(x) = 0 elsewhere.
Find the value of k.
Step by step
- The total area must be 1: integrate kx from 0 to 4 and set equal to 1.
- Integrate: the antiderivative of kx is k·x²/2.
- Evaluate at the limits.
- Solve for k.
Final answer
k = 1/8 = 0.125.
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IB-style question — k with a quadratic pdf
A continuous random variable X has pdf f(x) = kx² for 0 ≤ x ≤ 3, and 0 elsewhere.
Find k.
Step by step
- Set the total area to 1.
- Integrate: antiderivative of kx² is k·x³/3.
- Evaluate.
- Solve.
Final answer
k = 1/9.