2.1.3 Distribution function and Probability density function
The cumulative distribution function (CDF) or distribution function, 𝑭(𝒙), of the
random variable X is the probability that X takes a value less are equal to 𝑥. That is
𝐹 𝑥 = P(X ≤ 𝑥)
We write X~𝐹 to indicate that 𝐹 is the distribution function of X.
All probabilities about X can be found using its CDF.
For example, we can use Axiom 3 to find P(𝑎 < X ≤ 𝑏).
P X ≤ 𝑏 = P X ≤ 𝑎 + P(𝑎 < X ≤ 𝑏) Mutually exclusive eve
P 𝑎 < X ≤ 𝑏 = P X ≤ 𝑏 − P X ≤ 𝑎 = 𝐹 𝑏 − 𝐹(𝑎)
The cumulative distribution function (CDF) or distribution function, 𝑭(𝒙), of the
random variable X is the probability that X takes a value less are equal to 𝑥. That is
𝐹 𝑥 = P(X ≤ 𝑥)
We write X~𝐹 to indicate that 𝐹 is the distribution function of X.
All probabilities about X can be found using its CDF.
For example, we can use Axiom 3 to find P(𝑎 < X ≤ 𝑏).
P X ≤ 𝑏 = P X ≤ 𝑎 + P(𝑎 < X ≤ 𝑏) Mutually exclusive eve
P 𝑎 < X ≤ 𝑏 = P X ≤ 𝑏 − P X ≤ 𝑎 = 𝐹 𝑏 − 𝐹(𝑎)
