Part (a): Proving $2X$ and $3Y$ are Independent

To prove that $2X$ and $3Y$ are independent, we use the fact that $X$ and $Y$ are independent random variables. Given that $2X$ is just a linear transformation of $X$ and $3Y$ is a linear transformation of $Y$, these two linear transformations will also be independent.

Reasoning:

Two random variables $A$ and $B$ are independent if and only if the joint distribution of $A$ and $B$ can be factored into the product of their marginal distributions, i.e.,

$P(A \leq a, B \leq b) = P(A \leq a) \cdot P(B \leq b) \quad \text{for all } a, b.$

Since $2X$ and $3Y$ are obtained by multiplying independent random variables $X$ and $Y$ by constants, we know that for independent random variables $X$ and $Y$:

$P(2X \leq a, 3Y \leq b) = P\left(X \leq \frac{a}{2}\right) \cdot P\left(Y \leq \frac{b}{3}\right),$

which is equivalent to the product of the marginal distributions of $2X$ and $3Y$. Therefore, $2X$ and $3Y$ are independent.

Part (b): Standard Deviation of $2X + 3Y$

We are given:

$\text{Var}(X) = 5, \quad \text{Var}(Y) = 9.$

We need to find the standard deviation of $2X + 3Y$.

Step 1: Calculate the Variance of $2X + 3Y$

Since $X$ and $Y$ are independent, the variance of a linear combination of independent variables is given by:

$\text{Var}(2X + 3Y) = \text{Var}(2X) + \text{Var}(3Y).$

Now, using the property that if $a$ is a constant, $\text{Var}(aX) = a^2 \cdot \text{Var}(X)$, we have:

$\text{Var}(2X) = 2^2 \cdot \text{Var}(X) = 4 \cdot 5 = 20,$

$\text{Var}(3Y) = 3^2 \cdot \text{Var}(Y) = 9 \cdot 9 = 81.$

Thus,

$\text{Var}(2X + 3Y) = 20 + 81 = 101.$

Step 2: Calculate the Standard Deviation of $2X + 3Y$

The standard deviation is the square root of the variance:

$\text{SD}(2X + 3Y) = \sqrt{\text{Var}(2X + 3Y)} = \sqrt{101}.$

Final Answer:

• Part (a): $2X$ and $3Y$ are independent because they are linear transformations of independent variables $X$ and $Y$.
• Part (b): The standard deviation of $2X + 3Y$ is $\sqrt{101}$.

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Do you have any questions or would you like more details on any of the steps?

Related Questions:

1. What does it mean for two random variables to be independent?
2. How does a linear transformation affect the independence of random variables?
3. What is the relationship between variance and standard deviation?
4. Can you provide an example where two variables are not independent, and how their variance would be calculated differently?
5. What are the properties of variance when dealing with linear combinations of random variables?

Tip:

When dealing with the variance of a sum of independent random variables, always remember that the covariance term vanishes (since covariance between independent variables is zero), simplifying the calculation to the sum of individual variances.