Suppose that 𝑋 and 𝑌 are two independent random variables that are both distributed
𝒩(𝜇, 𝜎2
). Find the following: 𝔼(2𝑋 + 7𝑌 − 10)

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!

Suppose that 𝑋 and 𝑌 are two independent random variables that are both distributed 𝒩(𝜇, 𝜎^2). Find the following: 𝔼(2𝑋 + 7𝑌 − 10).

Find the expected value of the expression 2X + 7Y - 10, where X and Y are independent normal variables with mean μ and variance σ^2. Learn how to apply the linearity of expectation to solve this step-by-step.

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