If M = 4X + 4 then the E(M) is: If K = 2X + 2Y then the Var(K) is:

Understand the Problem

The questions are asking for the expected value of a random variable M defined in terms of X, and the variance of another random variable K defined in terms of X and Y. Specifically, it involves using properties of expected value and variance for independent random variables.

Answer

11. $20$ 12. $272$

Steps to Solve

Finding the Expected Value of M

To find the expected value $E(M)$ when $M = 4X + 4$, we use the property of expected value that states $E(aX + b) = aE(X) + b$, where $a$ is a constant and $b$ is a constant shift.

Here, $E(X) = 4$, so:

$$ E(M) = 4E(X) + 4 = 4(4) + 4 $$

Calculating this gives:

$$ E(M) = 16 + 4 = 20 $$

Finding the Variance of K

Next, we need to find the variance of $K = 2X + 2Y$. The variance of a linear combination of independent random variables can be calculated using the formula:

$$ Var(aX + bY) = a^2Var(X) + b^2Var(Y) $$

Here, $a = 2$ and $b = 2$, and we know that the variances are calculated as the square of the standard deviations:

$Var(X) = (2)^2 = 4$
$Var(Y) = (8)^2 = 64$

Now, plug in these values into the formula:

$$ Var(K) = 2^2Var(X) + 2^2Var(Y) = 4(4) + 4(64) $$

Calculating this gives:

$$ Var(K) = 16 + 256 = 272 $$

The answers are:

$E(M) = 20$
$Var(K) = 272$

More Information

In these problems, we applied the properties of expected value and variance to independent random variables. The first calculation shows how to combine the mean of X with a linear transformation, while the second calculation demonstrates how variance behaves with independent variables and scaling.

Tips

Forgetting to apply the properties of expected value and variance correctly, especially the scaling factors.
Confusing standard deviation and variance. Remember that variance is the square of the standard deviation.

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