Question

The following function is probability mass function.

x | -2 | -1 | 0 | 1 | 2
f(x) | 1/9 | 2/9 | 3/9 | 2/9| 1/9
Determine the mean and variance of the random variable. Give an exact answer in the form of a fraction.

Answer 1

The mean of the probability mass function is 0, and the variance is 4/3.

Step-by-step explanation:

To determine the mean and variance of the given probability mass function (PMF), we follow these steps:

1. Calculate the mean (also known as the expected value) by multiplying each value of the random variable (x) by its corresponding probability (f(x)) and then summing up all these products.
2. For the variance, we calculate the expected value of the squared deviations from the mean (E[(X - μ)^2]), which means we square each value of x, multiply by its probability, and sum these values up. Then we subtract the square of the mean (μ^2).

The calculation for the mean is as follows:

E[X] = (-2)*(1/9) + (-1)*(2/9) + (0)*(3/9) + (1)*(2/9) + (2)*(1/9) = -2/9 + -2/9 + 0 + 2/9 + 2/9 = 0

So the mean (μ) is 0.

The calculation for the variance is:

Variance = E[X^2] - (μ)^2 = ((-2)^2*(1/9) + (-1)^2*(2/9) + (0)^2*(3/9) + (1)^2*(2/9) + (2)^2*(1/9)) - 0^2

Variance = (4*(1/9) + 1*(2/9) + 0*(3/9) + 1*(2/9) + 4*(1/9)) - 0

Variance = (4/9 + 2/9 + 0 + 2/9 + 4/9) = 12/9 = 4/3

Therefore, the variance of the random variable is 4/3.