Probability Distributions and Probability Densities
1 Random Variables
2 Discrete Probability Distributions
3 Continuous Random Variables
1 Random Variables
In all of the examples of this section, we have limited our discussion to discrete sample space, and hence to a discrete random variable, namely, random variables whose range is finite or countably infinite. Continuous random variables defined over continuous sample spaces will be taken up in the third section.
2 Discrete Probability Distributions
3 Continuous Random Variables
So far we have considered discrete random variables that can take
on a finite or countably infinite number of values. In applications,
we are often interested in random variables that can take on an
uncountable continuum of values; we call these continuous random
variables.
Consider modeling the distribution of the age that a person dies at.
Age of death, measured perfectly with all the decimals and no
rounding, is a continuous random variable (e.g., age of death could
be 87.3248583585642 years). Other examples of continuous
random variables include: time until the occurrence of the next
earthquake in ˙
Istanbul; the lifetime of a battery; the annual rainfall
in Ankara.
Because it can take on so many different values, each value of a
continuous random variable winds up having probability zero. If I
ask you to guess someone’s age of death perfectly, not
approximately to the nearest millionth year, but rather exactly to
all the decimals, there is no way to guess correctly - each value
with all decimals has probability zero. But for an interval, say the
nearest half year, there is a nonzero chance you can guess
correctly. So, we have the following definition:
