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. $272$
Answer for screen readers
The answers are:
-
$E(M) = 20$
-
$Var(K) = 272$
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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