Cumulative Distribution Function

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 6 + 1 6 = 1 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 6 + 1 6 = 1 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 6 + 1 6 = 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

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 =0.03+0.04=0.07 𝐹 2 =𝑃 𝑋≤2 =0.03+0.04+0.06=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

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 =0.67−0.32 =0.35 S1: Page 158 8B (b) 𝑃 𝑋>2 =1−𝑃 𝑋≤2 =1−𝐹 2 =1−0.32 =0.87