Cumulative probabilities

The cumulative probability for any value \(x\) is the probability that the random variable's value is less than or equal to \(x\). This is the sum of the probabilities for \(x\) and all values below it.

\[P(X \le x) = \sum_{u \le x} p(u)\]

When these cumulative probabilities are considered together, they form the cumulative distribution function for \(X\).

Definition

The cumulative distribution function (CDF) for \(X\) is the function

\[F(x) = P(X \le x) = \sum_{u \le x} p(u)\]

The cumulative distribution function is a step function, satisfying

\[F(-\infty) = 0\\ F(+\infty) = 1\]

and increasing by \(p(x)\) at each \(x\).

Example

A couple want at least two children and no more than four, but will stop when they get a boy. Assuming that the probability of each child being a girl is \(\frac {1} {2} \), independently of the genders of previous children, the probability function for the number of girls in the resulting family is

Number of girls, x 0 1 2 3 4
p(x) 0.25 0.5 0.125 0.0625 0.0625

Draw the cumulative distribution function for X.

\(F(x)\) increases by \(p(x)\) at each x,

We now show an example based on a real data set.

Hurricanes in North Atlantic

The table below shows the numbers of hurricanes in the North Atlantic each year from 1910 to 2009.

Decade Year
Beginning 0 1 2 3 4 5 6 7 8 9
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
3
4
2
4
11
4
5
9
8
8
3
4
2
4
8
8
6
7
4
9
4
2
6
4
6
3
3
2
4
4
3
3
9
5
6
7
4
3
4
7
0
5
6
7
6
6
4
5
3
9
4
1
5
5
9
4
6
7
11
15
11
8
7
3
4
7
6
4
9
5
2
4
3
5
3
6
5
3
3
5
3
4
3
6
7
5
5
5
10
8
1
3
3
7
7
12
5
7
8
3

The diagram below displays the data in a jittered dot plot. It also shows the cumulative distribution function for the number of hurricanes in a year that has been randomly selected from the hundred.

Drag the vertical red line horizontally — the cumulative count increases in steps whose heights are the number of years that had each possible number of hurricanes. Note that the cumulative count remains constant between the integer values for the number of hurricanes.

Select Cumulative proportion to change the scale on the vertical axis to show the cumulative distribution function for the hurricane data.

These cumulative proportions are probabilities for a randomly selected year from the hundred in the data set. They will also be approximate probabilities for the number of hurricanes in any future year (assuming that there is no climate change).