Suppose that 𝑌 has the cumulative distribution function (cdf) Use the cdf to determine the 90th percentile of Y.

To determine the 90th percentile of a random variable $Y$ using its cumulative distribution function (CDF), follow these steps:

1. Break Down the Problem

We want to find the value $y$ such that the CDF $F_Y(y) = 0.90$. This value represents the point below which 90% of the data falls.

2. Relevant Concepts

The cumulative distribution function (CDF) $F_Y(y)$ is defined as: $F_Y(y) = P(Y \leq y)$ To find the 90th percentile, we will set $F_Y(y) = 0.90$ and solve for $y$.

3. Analysis and Detail

Assuming we have a specific form for the CDF $F_Y(y)$, we would solve the equation $F_Y(y) = 0.90$.

• For example, if the CDF is given as $F_Y(y) = \begin{cases} 0 & \text{if } y < a \\ \dots & \text{for other segments} \\ 1 & \text{if } y \geq b \end{cases}$
• You would look for the value of $y$ that satisfies this equation in the region between $a$ and $b$.

4. Verify and Summarize

After solving for the value of $y$ where the CDF equals 0.90, verify that when we substitute $y$ back into the CDF equation, it indeed equals 0.90.

Final Answer

The final result for the 90th percentile of $Y$ will be the specific value $y$ found from $F_Y(y) = 0.90$. Without the exact CDF, we cannot determine the numeric value. Please provide the CDF function for precise calculations.