A random variable X has an expected value of 10 and a standard deviation of 3. Let Y = 2X + 2. What are the expected value and standard deviation of Y?

To determine the expected value and standard deviation of \( Y = 2X + 2 \), we can use the properties of expected values and standard deviations for linear transformations of random variables.

### Expected Value of \( Y \)

The expected value of \( Y \) is given by:

\[
E(Y) = E(2X + 2)
\]

Using the linearity of expectation, we get:

\[
E(Y) = 2 \cdot E(X) + 2
\]

Given \( E(X) = 10 \), we substitute:

\[
E(Y) = 2 \cdot 10 + 2 = 20 + 2 = 22
\]

So, the expected value of \( Y \) is **22**.

### Standard Deviation of \( Y \)

The standard deviation of \( Y \) depends only on the multiplicative factor \( 2 \), since adding a constant does not change the spread. The formula for the standard deviation of \( Y = aX + b \) is:

\[
\text{Std}(Y) = |a| \cdot \text{Std}(X)
\]

Here, \( a = 2 \) and \( \text{Std}(X) = 3 \). So,

\[
\text{Std}(Y) = 2 \cdot 3 = 6
\]

Thus, the standard deviation of \( Y \) is **6**.

### Summary

- **Expected value of \( Y \):** 22
- **Standard deviation of \( Y \):** 6

Would you like further clarification on any of these steps?

---

1. What is the variance of \( Y \) given the variance of \( X \)?
2. How would the expected value and standard deviation change if \( Y = 3X + 5 \)?
3. What are the properties of expected value and standard deviation for linear transformations?
4. How would the calculations change if the transformation was non-linear?
5. How do these concepts apply in practical statistical problems?

**Tip:** When dealing with linear transformations of random variables, remember that adding constants only affects the mean, not the standard deviation.

Learn how to find the expected value and standard deviation of Y = 2X + 2 given that X has an expected value of 10 and a standard deviation of 3. This problem demonstrates properties of linear transformations of random variables in statistics, ideal for students in Grades 10-12.

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