To find $\mathbb{E}(2X + 7Y - 10)$, where $X$ and $Y$ are independent random variables both distributed as $\mathcal{N}(\mu, \sigma^2)$, we can use the linearity of expectation.

The linearity of expectation states that for any random variables $X$ and $Y$, and constants $a$, $b$, and $c$, the following holds:

$\mathbb{E}(aX + bY + c) = a\mathbb{E}(X) + b\mathbb{E}(Y) + c$

Step-by-step solution:

Given:

• $X \sim \mathcal{N}(\mu, \sigma^2)$
• $Y \sim \mathcal{N}(\mu, \sigma^2)$
• $X$ and $Y$ are independent random variables
• We are tasked with finding $\mathbb{E}(2X + 7Y - 10)$.

Using the linearity of expectation:

$\mathbb{E}(2X + 7Y - 10) = 2\mathbb{E}(X) + 7\mathbb{E}(Y) - 10$

Since both $X$ and $Y$ are normally distributed with mean $\mu$, we have:

$\mathbb{E}(X) = \mu \quad \text{and} \quad \mathbb{E}(Y) = \mu$

Substitute these into the equation:

$\mathbb{E}(2X + 7Y - 10) = 2\mu + 7\mu - 10$

Simplify:

$\mathbb{E}(2X + 7Y - 10) = 9\mu - 10$

Final Answer:

$\mathbb{E}(2X + 7Y - 10) = 9\mu - 10$

Let me know if you'd like further details, or have any questions.

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Here are 5 related questions to explore this topic further:

1. What is the variance of $2X + 7Y - 10$ given the same conditions?
2. How does the expectation change if $X$ and $Y$ are not independent?
3. How can the covariance of $2X + 7Y$ be computed if $X$ and $Y$ are correlated?
4. What would be the expectation if $X \sim \mathcal{N}(0, \sigma^2)$ and $Y \sim \mathcal{N}(1, \sigma^2)$?
5. How would the result change if the expression involved a non-linear transformation, like $X^2$ or $e^X$?

Tip:

Always remember that the linearity of expectation holds regardless of whether the random variables are independent or dependent!