Distribution function and its properties

Distribution function and its properties

We get the probability of a given event at a particular point. If we want to have the probability upto the point we get the probability P(X ≤ x). This type of probability is known as probability mass function. We can also find how the probability is distributed within certain limits. [P (X < x) or P (X > x) or P (a < x < b)].

Distribution Function for discrete random variable

Definition: Let X be a random variable ,the cumulative distribution function (c.d.f) of a random variable X is defined as F(x) = P(X≤x), 6x. It is called simply as distribution function.

Properties:

Distribution Function for continuous random variable

Properties

Example 9.12

A random variable X has the following probability mass function

i. Find the value of ‘a ’

ii. Find the c.d.f F(x) of X

iii. Evaluate : (a) P(X  ≥ 4) (b) P(X < 5) (c) P(3 ≤ X ≤6)

iv. P(X = 5) using F(x)

Solution:

Example 9.13

Let X be a random variable with p.d.f

Solution: