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Lesson Example Discussion Quiz: Class Homework

Lesson

Title: Mean and variance of discrete random variable

Grade: 9-a Lesson: S4-L2

Explanation: Hello students, let us learn a new topic in statistics today with definitions, concepts, examples, and worksheets included.

Lesson:

Definition: Discrete random variable:

A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,…….

A probability mass function is used to describe the probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values.

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Explanation:

Suppose a random variable X may take n different values, with the probability that \$X = x_i\$ defined to be \$P(X = x_i) = p_i\$.

The probabilities \$p_i\$ must satisfy the following:

1: Each probability is between zero and one

2: Sum of the probabilities is one

Definition: Mean of discrete random variable:

The mean or expected value of a random variable can also be defined as the weighted average of all the values of the variable.It is denoted by E(X) or \$\mu\$.

\$P(X = x_i)\$ is the probability mass function

$2$

Explanation:

Let 'X' be a random variable with \$x_1,x_2,x_3,.....,x_n\$ occuring with probabilities \$p_1,p_2,p_3,..p_n\$.

The mean of random variable 'X' or expectation of 'X' is

E(X) = \$x_1p_1 + x_2p_2 + x_3p_3 +.....+x_np_n = \sum_{i=1}^n x_ip_i\$

Definition: Variance of discrete random variable:

The variance of a random variable can be defined as the expected value of the square of the difference of the random variable from the mean.It is denoted by \$\sigma^2\$ or Var(X)

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Explanation: Let 'X' be a random variable with \$x_1,x_2,x_3,.....,x_n\$ occuring with probabilities \$p_1,p_2,p_3,..p_n\$.

Var(X) = \$\sum_{i=1}^n (x_i - \mu)^2 p_i\$

\$\sigma^2 = \sum_{i=1}^n (x_i)^2 p_i + (\mu^2) \sum_{i=1}^n p_i - 2 \mu \sum_{i=1}^n x_ip_i\$

Here,\$\sum_{i=1}^n x_ip_i = \mu,\sum_{i=1}^n p_i = 1\$

\$\sigma^2 = \sum_{i=1}^n (x_i)^2 p_i + \mu^2 - 2 \mu^2\$

\$\sigma^2 = \sum_{i=1}^n (x_i)^2 p_i - (\mu)^2\$

\$VAR(X) = E(X^2) - (E(X))^2\$

Where,\$E(X^2) = \sum_{i=1}^n (x_i)^2 p_i, E(X) = \mu\$