To solve this problem, we will use the concept of order statistics. Since $Y_1, Y_2, Y_3$ are an i.i.d. sample from the uniform distribution on $(0,1)$, we know:

1. The sample median for $n = 3$ is the second order statistic, $Y_{(2)}$.
2. The probability density function of $Y_{(2)}$, the second order statistic, is derived using the formula for order statistics. For the $k$-th order statistic of $n$ i.i.d. samples, the probability density function is given by: [ f_{(k)}(y) = \frac{n!}{(k-1)!(n-k)!} \cdot [F(y)]^{k-1} \cdot [1-F(y)]^{n-k} \cdot f(y), ] where $f(y)$ is the density of the original distribution and $F(y)$ is the cumulative distribution function (CDF).

Steps to Find $f_{(2)}(y)$:

• Here, $n = 3$, $k = 2$, and $f(y) = 1$ because the uniform distribution on $(0,1)$ has a constant PDF of $1$ on $(0,1)$. The CDF is $F(y) = y$ for $y \in (0,1)$.

• Plugging these into the formula: [ f_{(2)}(y) = \frac{3!}{(2-1)!(3-2)!} \cdot [F(y)]^{2-1} \cdot [1-F(y)]^{3-2} \cdot f(y). ]

• Simplify the constants: [ f_{(2)}(y) = 3 \cdot [F(y)] \cdot [1-F(y)]. ]

• Substituting $F(y) = y$: $f_{(2)}(y) = 3 \cdot y \cdot (1-y), \quad 0 < y < 1.$

Thus, the probability density function of the sample median $Y_{(2)}$ is: $f_{(2)}(y) = 3y(1-y), \quad 0 < y < 1.$

Explanation:

• This density function describes the likelihood of the sample median $Y_{(2)}$ taking on a specific value $y$ within the interval $(0,1)$. It is symmetric about $y = 0.5$, which makes sense given the uniform distribution.

Would you like a further breakdown or visual graph of this distribution? Let me know! 😊

Related Questions:

1. What is the expected value $E[Y_{(2)}]$ of the sample median $Y_{(2)}$?
2. How does the distribution of $Y_{(2)}$ change if the sample size increases to $n = 5$?
3. Can you derive the variance $\text{Var}(Y_{(2)})$ using this density function?
4. What happens to the symmetry of the distribution if the original distribution is not uniform but skewed?
5. How do higher-order statistics (e.g., $Y_{(3)}$) compare to $Y_{(2)}$ in terms of their density?

Tip:

Always check the limits of integration and ensure that the PDF integrates to 1 when working with derived distributions like this!