(i) Discrete random variable

(a) (i) Discrete random variable

A random variable whose set of possible values is either finite or countably infinite is called discrete random variable.

Example

1. Let X represent the sum of the numbers on the 2 dice, when two dice Tera are thrown. In this case the random variable X takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. So X is a discrete random variable.

2. Number of transmitted bits received in error.

(ii) Probability mass function

If X is a discrete random variable, then the function P (x) = P[X=x] is called the probability mass function of X.

(iii) Probability distribution

The values assumed by the random variable X presented with corresponding probabilities is known as the probability distribution of X.

(iv) Cumulative distribution or Distribution function of X [for discrete R.V]

The cumulative distribution function F(x) of a discrete random variable X with probability distribution P (x) is given by

(v) Properties of distribution function :

1. F(- ∞) = 0

2. F(∞) = 1

3. 0 ≤ F(x) ≤ 1

4. F(x₁) ≤ F(x₂) if x₁ < x₂

5. P(x₁ < X ≤ x₂) = F(x₂) - F (x₁)

6. P(x₁ ≤ X ≤  x₂) = F (x₂) - F (x₁) + P(X = x₁]

7. P(x₁ < X < x₂) = F (x₂) - F (x₁) - P (X = x₂]

8. P(x₁ ≤ X < x₂) = F(x₂) -F (x₁) - P (X = x₂) + P(X = x₁)

(vi) Results :

1. P(X ≤ ∞) = 1

2. P(X ≤ - ∞) = 0

3. If x₁ ≤ x₂ then P (X = x₁) ≤ P (X = x₂)

4. P(X > x) = 1 - P [X ≤ x]

5. P(X ≤ x) = 1 − P(X > x)

(vii) Expected value of a Discrete random variable X.

M Let X be a discrete random variable assuming values x₁, x₂, ..., x_n with corresponding probabilities P₁, P₂, ..., P_n. Then

is called the Expected value of X.

E(X) is also called commonly the mean or the expectation of X.

A useful identity states that for a function g,

(viii) The variance of a random varaible X.

It is defined by Var(X) = E[X - E (X)]²The variance, which is equal to the expected value of the square of the difference between X and its expected value. It is a measure of the spread of the possible values of X.

A useful identity is that no

Var(X) E[X2] - [E (X)]²

The quantity √Var (X) is called the standard deviation of X.