Cumulative Probabilities Example
Assume we have a random variable X. Cumulative probabilities provide, for each value x, the probability of a result less than or equal to X, P[ X <= x ]. ( <= is "less than or equal to." Don't blame me. . .call Netscape & Microsoft.)
Here's the probability distribution for a discrete random variable X
The cumulative distribution function tables, for each value x = 1, 2, 3, 4, the probability of a result less than or equal to x. (In efficient English "less than or equal to" is often written "at most" or "no more than." These three phrases have the same meaning.) For example
These probabilities can be tabled
Suppose that P[ X <= 3 ] = 0.65, P[ X <= 4 ] = 0.80 and no values between 3 and 4 are possible. Then P[ X = 4 ] = 0.80 - 0.65 = 0.15. So, to compute f(x), take
where x* is the largest possible value below x. If x is already the smallest value then
Suppose the number of finish flaws on an automobile has the following cumulative probabilities.
Answers in red!
It is far better to understand what information is in the table-what the table "means"-and to think about the question, than it is to try to memorize 5 different strategies for handling these cases.There is an "easy" (if tedious) method for recovering f(x)-from each cumulative probability subtract its predecessor. For this example:
You can use this table to find answers to the questions above. Whichever way you do it, your results should be the same.
Do you understand it? If so, perhaps you'd like to try the worksheet on this topic?