If the probability density function of a random variable X is `f(x)=x/2` in `0 le x le 2`, then `P(X gt 1.5 | X gt 1)` is equal to

Options

A. `7/16`
B. `3/4`
C. `7/12`
D. `21/64`

Answer

To solve the problem, we need to find the conditional probability \( P(X > 1.5 | X > 1) \) given the probability density function (PDF) \( f(x) = \frac{x}{2} \) for \( 0 \leq x \leq 2 \).

### Step-by-Step Solution:

1. **Identify the Events**:
- Let \( A \) be the event \( X > 1.5 \).
- Let \( B \) be the event \( X > 1 \).

We need to find \( P(A | B) \), which can be calculated using the formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

2. **Find \( P(A \cap B) \)**:
- The event \( A \cap B \) corresponds to \( X > 1.5 \), since if \( X > 1.5 \), it is also greater than 1.
- Therefore, \( P(A \cap B) = P(X > 1.5) \).

We calculate \( P(X > 1.5) \) using the PDF:
\[ P(X > 1.5) = \int_{1.5}^{2} f(x) \, dx = \int_{1.5}^{2} \frac{x}{2} \, dx \]

Now, we calculate the integral:
\[ = \frac{1}{2} \int_{1.5}^{2} x \, dx = \frac{1}{2} \left[ \frac{x^2}{2} \right]_{1.5}^{2} \]
\[ = \frac{1}{2} \left( \frac{2^2}{2} - \frac{(1.5)^2}{2} \right) = \frac{1}{2} \left( \frac{4}{2} - \frac{2.25}{2} \right) \]
\[ = \frac{1}{2} \left( 2 - 1.125 \right) = \frac{1}{2} \times 0.875 = \frac{0.875}{2} = \frac{7}{16} \]

3. **Find \( P(B) \)**:
- Now we calculate \( P(B) = P(X > 1) \):
\[ P(X > 1) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \frac{x}{2} \, dx \]
\[ = \frac{1}{2} \int_{1}^{2} x \, dx = \frac{1}{2} \left[ \frac{x^2}{2} \right]_{1}^{2} \]
\[ = \frac{1}{2} \left( \frac{2^2}{2} - \frac{1^2}{2} \right) = \frac{1}{2} \left( \frac{4}{2} - \frac{1}{2} \right) \]
\[ = \frac{1}{2} \left( 2 - 0.5 \right) = \frac{1}{2} \times 1.5 = \frac{3}{4} \]

4. **Calculate \( P(A | B) \)**:
Now we can substitute the values we found into the conditional probability formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{7}{16}}{\frac{3}{4}} = \frac{7}{16} \times \frac{4}{3} = \frac{7 \times 4}{16 \times 3} = \frac{28}{48} = \frac{7}{12} \]

### Final Answer:
Thus, \( P(X > 1.5 | X > 1) = \frac{7}{12} \).