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?

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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.