Consider the continuous random variable Y having p.d.f.
f(y) = 
3y
2
if 0 ≤ y ≤ 1
0 otherwise.
Find the mean and variance of Z =
√
Y

Given the continuous random variable \( Y \) with the probability density function (p.d.f.)

\[
f(y) = 
\begin{cases} 
3y^2 & \text{if } 0 \leq y \leq 1, \\
0 & \text{otherwise},
\end{cases}
\]

we are asked to find the mean and variance of \( Z = \sqrt{Y} \).

### Step 1: Finding the Mean of \( Z \)
The mean of \( Z \) is given by:

\[
E[Z] = E[\sqrt{Y}]
\]

The expected value of \( Z \) can be calculated by integrating the product of \( \sqrt{y} \) and the p.d.f. of \( Y \):

\[
E[Z] = \int_0^1 \sqrt{y} \cdot f(y) \, dy
\]

Substituting \( f(y) = 3y^2 \):

\[
E[Z] = \int_0^1 \sqrt{y} \cdot 3y^2 \, dy = 3 \int_0^1 y^{5/2} \, dy
\]

Now, calculate this integral:

\[
E[Z] = 3 \left[ \frac{y^{7/2}}{\frac{7}{2}} \right]_0^1 = 3 \cdot \frac{2}{7} = \frac{6}{7}
\]

So, the mean of \( Z \) is:

\[
E[Z] = \frac{6}{7}
\]

### Step 2: Finding the Variance of \( Z \)
The variance of \( Z \) is given by:

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

First, compute \( E[Z^2] \):

\[
E[Z^2] = E[Y] = \int_0^1 y \cdot f(y) \, dy
\]

Substituting \( f(y) = 3y^2 \):

\[
E[Z^2] = \int_0^1 y \cdot 3y^2 \, dy = 3 \int_0^1 y^3 \, dy
\]

Now, calculate this integral:

\[
E[Z^2] = 3 \left[ \frac{y^4}{4} \right]_0^1 = 3 \cdot \frac{1}{4} = \frac{3}{4}
\]

Thus,

\[
\text{Var}(Z) = E[Z^2] - (E[Z])^2 = \frac{3}{4} - \left(\frac{6}{7}\right)^2
\]

Calculating the variance:

\[
\left(\frac{6}{7}\right)^2 = \frac{36}{49}
\]

So,

\[
\text{Var}(Z) = \frac{3}{4} - \frac{36}{49}
\]

Convert both fractions to have a common denominator:

\[
\frac{3}{4} = \frac{147}{196}, \quad \frac{36}{49} = \frac{144}{196}
\]

Thus,

\[
\text{Var}(Z) = \frac{147}{196} - \frac{144}{196} = \frac{3}{196}
\]

Therefore, the variance of \( Z \) is:

\[
\text{Var}(Z) = \frac{3}{196}
\]

### Final Answers:
- **Mean of \( Z \)**: \( \frac{6}{7} \)
- **Variance of \( Z \)**: \( \frac{3}{196} \)

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the mean and variance change if \( f(y) = 4y^3 \) instead?
2. What is the cumulative distribution function (C.D.F.) of \( Z = \sqrt{Y} \)?
3. How do you compute the skewness of \( Z \)?
4. What is the interpretation of the mean and variance in the context of a real-world application?
5. How does the transformation \( Z = \sqrt{Y} \) affect the shape of the distribution?

**Tip:** Understanding transformations of random variables can help solve complex problems by simplifying the original variable's distribution.

Learn how to calculate the mean and variance of Z = √Y when Y follows a probability density function f(y) = 3y^2. Understand the concepts of expected value and variance in probability theory. Suitable for advanced undergraduate students studying probability and statistics.

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