Let X be a random variable with the following probability...

Let $X$ be a random variable with the following probability distribution:
$$
\begin{array}{c|ccc}
x & -3 & 6 & 9 \\
\hline f(x) & 1 / 6 & 1 / 2 & 1 / 3
\end{array}
$$
Find $\mu_{g(X)},$ where $g(X)=(2 X+1)^{2}$.

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Key Concepts

Discrete Random Variable

A discrete random variable is a variable that can take on a countable number of distinct values. Its behavior is fully described by a probability mass function (PMF), which assigns a probability to each value the variable can assume.

Probability Distribution

A probability distribution for a discrete random variable lists all possible outcomes along with their corresponding probabilities. It is fundamental to understanding the behavior of the random variable and is used as the basis for calculating moments and other statistical properties.

Mathematical Expectation

Mathematical expectation, or the expected value, is a measure of the central tendency of a random variable. It is calculated as a sum of all possible values weighted by their probabilities. For a function of a random variable, the expected value is computed by summing the product of the function’s value at each outcome and the corresponding probability.

Transformation of a Random Variable

A transformation of a random variable involves applying a function to the variable, which results in a new random variable. The properties of the new variable, including its expectation, can be determined by applying the transformation function to each outcome of the original variable and then using the original probabilities.

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