To find the variance of a discrete random variable [tex]\( X \)[/tex] with the given probability mass function (pmf), we need to follow these steps:
1. Determine the Mean (Expected Value) of [tex]\( X \)[/tex]:
The mean (expected value) [tex]\(\mu\)[/tex] of a discrete random variable [tex]\( X \)[/tex] is calculated using the formula:
[tex]\[
\mu = E[X] = \sum_{x} x \cdot P_X(x)
\][/tex]
Here, the random variable [tex]\( X \)[/tex] can take on the values 0 and 2 with probabilities [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex], respectively.
Therefore,
[tex]\[
\mu = 0 \cdot \frac{1}{3} + 2 \cdot \frac{2}{3} = 0 + \frac{4}{3} = \frac{4}{3}
\][/tex]
2. Calculate the Variance of [tex]\( X \)[/tex]:
The variance [tex]\(\text{Var}(X)\)[/tex] is calculated using the formula:
[tex]\[
\text{Var}(X) = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 \cdot P_X(x)
\][/tex]
Plugging in the values we have:
[tex]\[
\text{Var}(X) = (0 - \frac{4}{3})^2 \cdot \frac{1}{3} + (2 - \frac{4}{3})^2 \cdot \frac{2}{3}
\][/tex]
Simplifying each term individually:
For [tex]\( x = 0 \)[/tex]:
[tex]\[
(0 - \frac{4}{3})^2 \cdot \frac{1}{3} = (-\frac{4}{3})^2 \cdot \frac{1}{3} = \frac{16}{9} \cdot \frac{1}{3} = \frac{16}{27}
\][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[
(2 - \frac{4}{3})^2 \cdot \frac{2}{3} = (\frac{2}{3})^2 \cdot \frac{2}{3} = \frac{4}{9} \cdot \frac{2}{3} = \frac{8}{27}
\][/tex]
Adding these terms together, we get:
[tex]\[
\text{Var}(X) = \frac{16}{27} + \frac{8}{27} = \frac{24}{27} = \frac{8}{9}
\][/tex]
Therefore, the mean (expected value) of [tex]\( X \)[/tex] is [tex]\( \frac{4}{3} \)[/tex] and the variance of [tex]\( X \)[/tex] is [tex]\( \frac{8}{9} \)[/tex] or approximately [tex]\( 0.8888888888888888 \)[/tex].
