Lesson 26 of 1524
Continuous random variables
Probability is area. A single exact value has no width, so its probability is 0.
Practice this chapterA continuous variable is measured. The probability that it falls in an interval is the area under the density curve over that interval. The whole area is 1.
The uniform density is flat between two endpoints, so the probability is the length of your interval divided by the total length. The exponential distribution is the other model in this chapter, used for waiting times.
Uniform probability
P(c < X < d) = (d − c) / (b − a)
Length of the event, divided by length of the whole range.
Worked example
X is uniform from 0 to 10. What is P(2 < X < 5)?
- 1A uniform density is flat, so probability is the length of the event divided by the length of the whole range.
- 2The event runs from 2 to 5, so its length is 5 − 2 = 3.
- 3The whole range runs from 0 to 10, so its length is 10 − 0 = 10.
- 4P(2 < X < 5) = 3 / 10 = 0.3.
- 5A single point has no width. P(X = 2) = 0, and including the endpoints would not change this area.
Result: 0.3
Why. Three units of a ten-unit range are inside the event, and every unit of a uniform distribution is equally likely. That is why the probability is 3/10 rather than the height of the curve or the value 2.
P(X = 2) is 0 for a continuous variable. Endpoints do not change the area, so P(2 < X < 5) and P(2 ≤ X ≤ 5) are the same.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.