1. Cumulative Distribution Function

2. Cumulative Distribution Functions (c.d.f.)
If 𝑋 is a discrete r.v. we can find a cumulative probability by adding up all the probabilities up to a certain value. We denote the cumulative probability using 𝐹 𝑥 =𝑃(𝑋≤𝑥) Example: given the distribution for 𝑋 shown, find the cumulative distribution function x
P(X = )
𝐹 1 =𝑃 𝑋≤1 = 1 6
𝐹 2 =𝑃 𝑋≤2 =𝑃 𝑋=1 +𝑃 𝑋=2 = = 1 3

3. Cumulative Distribution Functions (c.d.f.)x
P(X = )
𝐹 1 =𝑃 𝑋≤1 = 1 6
𝐹 2 =𝑃 𝑋≤2 =𝑃 𝑋=1 +𝑃 𝑋=2 = = 1 3

4. Cumulative Distribution Functions (c.d.f.)x
P(X = )
𝐹 1 =𝑃 𝑋≤1 = 1 6
𝐹 2 =𝑃 𝑋≤2 =𝑃 𝑋=1 +𝑃 𝑋=2 = = 2 6
𝐹 3 =𝑃 𝑋≤3 = 3 6
𝐹 4 =𝑃 𝑋≤4 = 4 6
𝐹 5 =𝑃 𝑋≤5 = 5 6
𝐹 6 =𝑃 𝑋≤6 = 6 6
Therefore 𝐹 𝑥 = 𝑥 6 for 𝑥=1,2,3,…,6
It is not always possible to write a formula – see next example

5. It is not always possible to write a formula
Example 2
The probability distribution for the r.v. 𝑋 is shown in the table. Construct the cumulative distribution table. 𝑥 1 2 3 4 5 6
𝑃(𝑋=𝑥)
0.03
0.04
0.06
0.12
0.4
0.15
0.2
𝐹 0 =𝑃 𝑋≤0 =0.03
𝐹 1 =𝑃 𝑋≤1 = =0.07
𝐹 2 =𝑃 𝑋≤2 = =0.13
And so on. This gives us the cumulative distribution table
It is not always possible to write a formula 𝑥 1 2 3 4 5 6
𝐹(𝑥)
0.03
0.07
0.13
0.25
0.65
0.8

6. Example 3
Given the cumulative distribution function 𝐹(𝑥) for the discrete r.v. 𝑋, find (a) 𝑃 𝑋=3 (b) 𝑃(𝑋>2) 𝑥 1 2 3 4 5
𝐹(𝑥)
0.2
0.32
0.67
0.9
Solution (a) From the table
𝐹 3 =𝑃 𝑋≤3 =𝑃 𝑋=1 +𝑃 𝑋=2 +𝑃 𝑋=3 =0.67
𝐹 2 =𝑃 𝑋≤2 =𝑃 𝑋=1 +𝑃 𝑋=2 =0.32
Now 𝑃 𝑋=3 =𝐹 3 −𝐹 2
=0.67− =0.35
S1: Page 158 8B
(b) 𝑃 𝑋>2 =1−𝑃 𝑋≤2
=1−𝐹 2
=1−0.32
=0.87