Previously we studied discrete random variables. What characterized them was that their possible values are distinct and separate from each other.
One example is the sum of the results of throwing 2 dice:
In this lecture we will study about continuous random variables. That is, random variables whose possible values form an entire interval.
For example, the exact length of a fish, the exact time it takes you to complete a marathon, or the result of a perfect random number generator (which can return any value between 0 and 1).
In such cases it is not possible to assign a probability to each possible value. There are just two many possible values. Then, who do we express the probability of a continuous random variable?
To illustrate this, let us look at the exact time interval between eruptions of the Old Faithful geyser at Yellowstone Park, Wyoming. Over the years, thousands of eruptions have taken place, which comes in two flavours, long ones and short ones:
Then, how would you calculate the probability of a time interval between 60 and 90 minutes?
Suppose you round off to the nearest multiple of 10 minutes, and let $X$ be the length of a time interval. $X$ will then be a discrete random variable, with possible values:
As a result, $P(60 \leq X \leq 90)=p_{60}+p_{70}+p_{80}+p_{90}$
You can visualize this table by putting adjacent vertical bars around each possible value and taking the height in such a way that the area of each bar is equal to the corresponding probability:
Based on measurements collected over the years, we then we something like this:
The total area of all the bars is equal to 1 since it is the sum of all probabilities.