Download Probability Density Functions A continuous random variable takes

Probability Density Functions
A continuous random variable takes on an uncountably infinite number of possible
values. For a discrete random variable X that takes on a finite or countably infinite number of
possible values, we determined P(X = x) for all of the possible values of X, and called it the
probability mass function ("p.m.f."). For continuous random variables, as we shall soon see,
the probability that X takes on any particular value x is 0. That is, finding P(X = x) for a
continuous random variable X is not going to work. Instead, we'll need to find the probability
that X falls in some interval (a, b), that is, we'll need to find P(a < X < b). We'll do that using a
probability density function ("p.d.f."). We'll first motivate a p.d.f. with an example, and then
we'll formally define it.
Example
Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you
can well imagine that it is not exactly 0.25 pounds. One randomly selected hamburger might
weigh 0.23 pounds while another might weigh 0.27 pounds. What is the probability that a
randomly selected hamburger weighs between 0.20 and 0.30 pounds? That is, if we
let X denote the weight of a randomly selected quarter-pound hamburger in pounds, what
is P(0.20 < X < 0.30)?