## Cumulative Probabilities ExampleAssume we have a random variable ## Example 1Here's the probability distribution for a
discrete random variable
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 - P[ X <= 1 ] = 0.1
- P[ X <= 2 ] = 0.1 + 0.2 = 0.3
- P[ X <= 3 ] = 0.1 + 0.2 + 0.4 = 0.7
- P[ X <= 4 ] = 0.1 + 0.2 + 0.4 + 0.3 = 1
These probabilities can be tabled
## Example 2Suppose that P[
where
Suppose the number of finish flaws on an automobile has the following cumulative probabilities.
## QuestionsAnswers in red! - Find the probability there are at most 2 flaws,
P[
*X**<=*2 ]. 0.87949 - Find the probability there are exactly 2 flaws,
P[
*X*= 2 ]. 0.87949 - 0.66263 =**0.21686** - Find the probability there are less than 2 flaws,
P[
*X*< 2 ].
**0.66263** - Find the probability there are more than 2 flaws,
P[
*X*> 2 ]. 1 - 0.87949 =**0.12051** - Find the probability there are at least 2 flaws,
P[
*X*>= 2 ]. 1 - 0.66263 =**0.33737** - Find the probability there are no flaws.
P[*X*= 0 ] = P[*X*<= 0] =**0.30119** - Find the probability there is at least one flaw.
1 - 0.30119 =**0.69881** - Find the probability there are between 1 and 3
flaws, inclusive, P[ 1
*<= X**<=*3 ]. 0.96623 - 0.30119 =**0.66504**
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
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? |