Let E(Y | X = x) = 2x, Var(Y | X = x) = 4x², and let X have a uniform distribution on the interval (0, 1). What is Var(Y) ?

To find \(\text{Var}(Y)\), we can use the law of total variance. The law of total variance states:

\[
\text{Var}(Y) = \mathbb{E}[\text{Var}(Y | X)] + \text{Var}(\mathbb{E}[Y | X])
\]

Given:

- \(\mathbb{E}(Y | X = x) = 2x\)
- \(\text{Var}(Y | X = x) = 4x^2\)
- \(X\) has a uniform distribution on the interval \((0, 1)\), i.e., \(X \sim U(0, 1)\)

### Step 1: Calculate \(\mathbb{E}[\text{Var}(Y | X)]\)

\[
\mathbb{E}[\text{Var}(Y | X)] = \mathbb{E}[4X^2]
\]

Since \(X \sim U(0, 1)\), the expected value of \(X^2\) is:

\[
\mathbb{E}[X^2] = \int_0^1 x^2 \, dx
\]

Let's compute this integral:

\[
\mathbb{E}[X^2] = \int_0^1 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3}
\]

So,

\[
\mathbb{E}[\text{Var}(Y | X)] = 4 \cdot \mathbb{E}[X^2] = 4 \cdot \frac{1}{3} = \frac{4}{3}
\]

### Step 2: Calculate \(\text{Var}(\mathbb{E}[Y | X])\)

\[
\text{Var}(\mathbb{E}[Y | X]) = \text{Var}(2X)
\]

Since \(\mathbb{E}[Y | X] = 2X\), we calculate the variance of \(2X\). Because \(X \sim U(0, 1)\), the variance of \(X\) is:

\[
\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2
\]

We already know \(\mathbb{E}[X^2] = \frac{1}{3}\) and \(\mathbb{E}[X] = \frac{1}{2}\). Therefore:

\[
\text{Var}(X) = \frac{1}{3} - \left(\frac{1}{2}\right)^2 = \frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12}
\]

Now, since \(\text{Var}(2X) = 4 \cdot \text{Var}(X)\):

\[
\text{Var}(2X) = 4 \cdot \frac{1}{12} = \frac{4}{12} = \frac{1}{3}
\]

### Step 3: Calculate \(\text{Var}(Y)\)

Finally, we combine these results using the law of total variance:

\[
\text{Var}(Y) = \mathbb{E}[\text{Var}(Y | X)] + \text{Var}(\mathbb{E}[Y | X]) = \frac{4}{3} + \frac{1}{3} = \frac{5}{3}
\]

So, \(\text{Var}(Y) = \frac{5}{3}\).

Would you like more details or have any questions?

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**Related Questions:**

1. How does the law of total variance apply to other distributions?
2. What if \(X\) had a different distribution, such as normal or exponential? How would that change \(\text{Var}(Y)\)?
3. How do we compute higher moments like \(\text{Var}(\text{Var}(Y|X))\)?
4. Can \(\text{Var}(Y)\) ever be less than \(\mathbb{E}[\text{Var}(Y|X)]\)?
5. How would the result change if \(\text{Var}(Y|X) = \text{constant}\)?

**Tip:** Always double-check your use of expectation and variance identities, especially when dealing with conditional distributions.

Learn how to calculate the variance of Y given its conditional expectation and variance, under a uniform distribution of X. Apply the law of total variance to find the precise value. Suitable for advanced undergraduate students and researchers in probability theory and statistics.

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