Lectures on Elementary Probability

Chapter 5
Continuous Random
Variables
5.1 Mean
A continuous random variable X with a probability density function f is a random
variable such that for each interval B of real numbers
∫
P [X ∈ B] = f(x) dx. (5.1)
Such a continuous random variable has the property that for each real number
x the event that X = x has a probability zero. This does not contradict the
countable additivity axiom of probability theory, since the set of real number
cannot be arranged in a sequence.
Often the random variable X has units, such as seconds or meters. In such a
case the probability density function f has values f(x) that have inverse units:
inverse seconds or inverse centimeters. This is so that the integral that defines
the probability will be dimensionless. The inverse dimensions of f(x) and the
dimensions of dx cancel, and the final integral is a dimensionless probability.
The values of a probability density function are not probabilities. One goes
from the probability density function to probability by integration:
P [X ≤ b] =
∫ b
B
−∞
f(x) dx. (5.2)
In the other direction, one goes from probabilities to the probability density
function by differentiation:
f(x) = d P [X ≤ x]. (5.3)
dx
In spite of this somewhat subtle relation, most of our thinking about continuous
random variables involves their probability density functions.
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