Questions: For a uniform random variable U ~ U(10,19), find u0.75: u0.75= (Round the answer to 2 decimal places)

Transcript text: For a uniform random variable $U \sim U(10,19)$, find $u_{0.75}:$ $u_{0.75}=$ $\square$ (Round the answer to 2 decimal places)

Solution

Step 1: Define the Uniform Distribution

We are given a uniform random variable $ U \sim U(10, 19) $. The parameters of this distribution are:

Step 2: Calculate the 0.75 Quantile

The quantile function for a uniform distribution is defined as:

\[ Q(p) = a + p \times (b - a) \]

For our case, we want to find the 0.75 quantile, so we set $ p = 0.75 $:

\[ Q(0.75) = 10 + 0.75 \times (19 - 10) \]

Calculating the difference:

\[ 19 - 10 = 9 \]

Now substituting back into the equation:

\[ Q(0.75) = 10 + 0.75 \times 9 \]

Calculating $ 0.75 \times 9 $:

\[ 0.75 \times 9 = 6.75 \]

Thus, we have:

\[ Q(0.75) = 10 + 6.75 = 16.75 \]

Step 3: Round the Result

The calculated quantile $ Q(0.75) = 16.75 $ is already rounded to two decimal places.

The 0.75 quantile of the uniform distribution $ U(10, 19) $ is:

\[ \boxed{16.75} \]

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