Math Snap

STEP 1

1. $X$ is a normally distributed random variable with mean $\mu = 9$ and variance $\sigma^2 = 25$.
2. We need to find the expected value of the expression $(5X - 4)^2$.
3. The properties of expectation and variance for linear transformations of random variables will be used.

STEP 2

1. Identify the transformation of the random variable.
2. Apply the properties of expectation.
3. Calculate the expected value.

STEP 3

Identify the transformation of the random variable $X$. We are given the expression $(5X - 4)^2$. This is a transformation of $X$ where $Y = 5X - 4$.

STEP 4

Apply the properties of expectation. We need to find $E(Y^2)$ where $Y = 5X - 4$. The expectation of a square of a linear transformation can be found using the formula:
$$E(Y^2) = \text{Var}(Y) + [E(Y)]^2$$

STEP 5

Calculate $E(Y)$ and $\text{Var}(Y)$:
1. $E(Y) = E(5X - 4) = 5E(X) - 4 = 5 \times 9 - 4 = 45 - 4 = 41$
2. $\text{Var}(Y) = \text{Var}(5X - 4) = 5^2 \text{Var}(X) = 25 \times 25 = 625$

Now, substitute these values into the formula for $E(Y^2)$:
$$E(Y^2) = \text{Var}(Y) + [E(Y)]^2 = 625 + 41^2$$ Calculate $41^2$:
$$41^2 = 1681$$ Therefore:
$$E(Y^2) = 625 + 1681 = 2306$$ The expected value $E(5X - 4)^2$ is $\boxed{2306}$.