How to calculate CDF when X is discontinued?

Question

recently I am doing a question

1. I want to find P(2 < x ≤ 3)

I can tell P(x ≤ 3) is 0.4 [P(5 ≤ x) + P(3 ≤ x < 5) = 0.2 +0.2)]

However, how can I get 2 < x?

2. I want to find P( x = 3)

However, when 3 ≤ x < 5, x can be 3 or 4. How can I find x = 3?

3. I want to find P( x = 6)

Is it equal to 0.2?

I know there may be lots of questions, but I cannot really clarify the concepts of it. Many thanks!!!

Answer 1

• $P(X \le 3)$ is actually just $F(3)$.
• $P(2 < X \le 3) = P(X \le 3) - P(X \le 2) = F(3) - F(2)$
• $P(X = 3) = P(X \le 3) - \lim_{t \uparrow 3} P(X \le t) = F(3) - \lim_{t \uparrow 3} F(t)$
• $P(X=6)=0$ because $P(X \le 5) = F(5)=1$.