A discript random variable x has the following distribution X/2/3/1 P(x)/0.5/0.3/0.2 Find the mean and then variance of g(x)=x squared and then used the example above to complete a linear function g(x)=2x+1

The mean of a discrete random variable is given by $E(X) = \sum x \cdot P(x)$.

$$\begin{align*} E(X) &= (1 \times 0.2) + (2 \times 0.5) + (3 \times 0.3) \\ &= 0.2 + 1.0 + 0.9 \\ &= 2.1 \end{align*}$$

The mean of $X$ is $\boxed{2.1}$.

First, we need to calculate $E(X^2) = \sum x^2 \cdot P(x)$.

$$\begin{align*} E(X^2) &= (1^2 \times 0.2) + (2^2 \times 0.5) + (3^2 \times 0.3) \\ &= (1 \times 0.2) + (4 \times 0.5) + (9 \times 0.3) \\ &= 0.2 + 2.0 + 2.7 \\ &= 4.9 \end{align*}$$

Now, we can calculate the variance using the formula $Var(X) = E(X^2) - (E(X))^2$.

$$\begin{align*} Var(X) &= 4.9 - (2.1)^2 \\ &= 4.9 - 4.41 \\ &= 0.49 \end{align*}$$

The variance of $X$ is $\boxed{0.49}$.

The mean of $g(X) = X^2$ is $E(X^2)$, which we already calculated in Step 2.

$$E(X^2) = 4.9$$

The mean of $g(X) = X^2$ is $\boxed{4.9}$.

To find $Var(X^2)$, we need $E((X^2)^2) = E(X^4)$.

$$\begin{align*} E(X^4) &= (1^4 \times 0.2) + (2^4 \times 0.5) + (3^4 \times 0.3) \\ &= (1 \times 0.2) + (16 \times 0.5) + (81 \times 0.3) \\ &= 0.2 + 8.0 + 24.3 \\ &= 32.5 \end{align*}$$

Now, we can calculate $Var(X^2) = E(X^4) - (E(X^2))^2$.

$$\begin{align*} Var(X^2) &= 32.5 - (4.9)^2 \\ &= 32.5 - 24.01 \\ &= 8.49 \end{align*}$$

The variance of $g(X) = X^2$ is $\boxed{8.49}$.

Using the property $E(aX+b) = aE(X)+b$:

$$\begin{align*} E(2X+1) &= 2E(X) + 1 \\ &= 2(2.1) + 1 \\ &= 4.2 + 1 \\ &= 5.2 \end{align*}$$

The mean of $g(X) = 2X+1$ is $\boxed{5.2}$.

Using the property $Var(aX+b) = a^2 Var(X)$:

$$\begin{align*} Var(2X+1) &= 2^2 Var(X) \\ &= 4 \times 0.49 \\ &= 1.96 \end{align*}$$

The variance of $g(X) = 2X+1$ is $\boxed{1.96}$.