# Cumulative Distribution Function (CDF)

## Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable is defined as the probability that the random variable X takes value less than or equal to x.

i.e., CDF F_{X}(x) = P(X ≤ x) …………….(1)

Here, x is a dummy variable and the CDF is denoted by F_{X}(x).

It is possible to define CDF for continuous as well as the discrete random variables. The CDF is sometimes called as simply the distribution function.

### Important Properties of CDF

**Property 1**: The CDF is always bounded between 0 and 1.

i.e., 0 ≤ F_{X}(x) ≥ 1 ………….(2)

As per the definition of CDF, it is a probability function P(X ≤ x) and any probability must have a value between 0 and 1. Therefore, CDF is always bounded between 0 and 1.

**Property 2:** This property states that,

F_{X}(∞)= 1 …………(3)

Proof: Here, F_{X}(∞) = P (X ≤ ∞). This includes the probability of all the possible outcomes or events. The random variable X ≤ ∞, thus, becomes a ‘certain event’ and therefore has a 100% probability.

**Property 3:** This property states that,

F_{X}(-∞) = 0 ……………..(6.18)

Proof: Here, F_{X}(-∞) = P(X ≤ -∞). The random variable X cannot have any value which is less than or equal to – ∞. Thus, X ≤ -∞ is a null event and therefore, has a 0% probability.

**Property 4:** This property states that F_{X}(x) is a monotone non-decreasing function i.e.,

F_{X}(x1) ≤ F_{X}(x2) for x1 < x2 ……………….(4)

Proof: To prove this property, let us consider fig.1.

Fig.1

F_{X}(x2) is defined as under: F_{X}(x2) = P(X ≤ x2)

The R.H.S of this equation can be expressed as union of two probabilities.

Therefore,

F_{X}(x2) = P(X ≤ x2) = P(X ≤ x1) ∪ P (x1 < X ≤ x2) ………………. (5)

The two events X ≤ x1 and x1 < X ≤ x2 are mutually exclusive.

Therefore, we write

P(X ≤ x1) ∪ P (x1 < X ≤ x2) = P(X ≤ x1) + P (x1 < X ≤ x2)

Substituting this value in equation (5), we get

F_{X}(x2) = P(X ≤ x1) + P(x1 < X ≤ x2) …………….(6)

But, the first term in equation (6) i.e., P(X ≤ x1) = F_{X}(x1)

Therefore,

F_{X}(x2) = F_{X}(x1) + P(x1 < X ≤ x2) ………………(7)

P(x1 < X ≤ x2) is always non-negative as it is a probability function, Thus,

F_{X}(x2) ≥ F_{X}(x1)

or F_{X}(x1) ≤ F_{X}(x2) for x1 ≤ x2

These properties of CDF are general and are valid for the continuous as well as discrete random variables.