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# Distribution function and its properties

i. Distribution Function for discrete random variable ii. Distribution Function for continuous random variable

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(Xx). 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(Xx), 6x. It is called simply as distribution function.

## 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)

### Example 9.13

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

### Solution:

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11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation : Distribution function and its properties |