To calculate $E[W]$, where $W = 3X - 5Y$, we use the linearity of expectation:

[ E[W] = E[3X - 5Y] = 3E[X] - 5E[Y] ]

Thus, we need to compute $E[X]$ and $E[Y]$ first.

Step 1: Compute $E[X]$

The expected value of $X$, $E[X]$, is given by:

$E[X] = \sum x \cdot p(x)$

Substitute the given values for $x$ and $p(x)$:

$E[X] = (0 \cdot 0.1) + (1 \cdot 0.5) + (2 \cdot 0.4)$ $E[X] = 0 + 0.5 + 0.8 = 1.3$

Step 2: Compute $E[Y]$

The expected value of $Y$, $E[Y]$, is given by:

$E[Y] = \sum y \cdot p(y)$

Substitute the given values for $y$ and $p(y)$:

$E[Y] = (10 \cdot 0.3) + (2 \cdot 0.1) + (8 \cdot 0.2) + (4 \cdot 0.4)$ $E[Y] = 3 + 0.2 + 1.6 + 1.6 = 6.4$

Step 3: Compute $E[W]$

Now, substitute the values of $E[X]$ and $E[Y]$ into the formula for $E[W]$:

[ E[W] = 3E[X] - 5E[Y] ] $E[W] = 3(1.3) - 5(6.4)$ $E[W] = 3.9 - 32 = -28.1$

Final Answer:

$E[W] = -28.1$

Would you like more details on any of these steps? Here are some related questions you might find helpful:

1. How do you calculate the variance of a linear combination of random variables like $W = 3X - 5Y$?
2. What is the relationship between covariance and the expectation of a linear combination of two random variables?
3. How can you compute the expected value for a discrete random variable with a probability distribution?
4. What are the key properties of expectation in probability theory?
5. How do you interpret the expected value in the context of a random variable?

Tip: When calculating expected values for linear combinations, remember the rule of linearity of expectation: $E[aX + bY] = aE[X] + bE[Y]$.