Let X be uniform over (0, 1). Find E[X|X < 1 2 ].

To solve \( E[X \mid X < \frac{1}{2}] \), where \( X \) is a random variable uniformly distributed over \( (0, 1) \), we can follow these steps:

### Step 1: Define the conditional expectation
The conditional expectation \( E[X \mid X < \frac{1}{2}] \) represents the expected value of \( X \) given that \( X \) lies within the interval \( (0, \frac{1}{2}) \). Since \( X \) is uniform over \( (0, 1) \), we know that the probability density function (pdf) of \( X \) is:
\[
f_X(x) = 1 \quad \text{for } 0 < x < 1.
\]

### Step 2: Find the new distribution over \( (0, \frac{1}{2}) \)
Given that \( X < \frac{1}{2} \), we now restrict \( X \) to the interval \( (0, \frac{1}{2}) \). The conditional pdf of \( X \) given \( X < \frac{1}{2} \) is:
\[
f_{X|X < \frac{1}{2}}(x) = \frac{f_X(x)}{\mathbb{P}(X < \frac{1}{2})} = \frac{1}{\frac{1}{2}} = 2 \quad \text{for } 0 < x < \frac{1}{2}.
\]
This is just the uniform distribution over \( (0, \frac{1}{2}) \).

### Step 3: Calculate the conditional expectation
The conditional expectation is given by:
\[
E[X \mid X < \frac{1}{2}] = \int_0^{\frac{1}{2}} x f_{X|X < \frac{1}{2}}(x) \, dx = \int_0^{\frac{1}{2}} x \cdot 2 \, dx.
\]
Evaluating the integral:
\[
E[X \mid X < \frac{1}{2}] = 2 \int_0^{\frac{1}{2}} x \, dx = 2 \left[ \frac{x^2}{2} \right]_0^{\frac{1}{2}} = 2 \left( \frac{\left(\frac{1}{2}\right)^2}{2} \right) = 2 \times \frac{1}{8} = \frac{1}{4}.
\]

### Final Answer:
\[
E[X \mid X < \frac{1}{2}] = \frac{1}{4}.
\]

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How do you compute conditional probability for uniform distributions?
2. What is the general formula for conditional expectation?
3. How does the uniform distribution over \( (0, 1) \) compare to other continuous distributions?
4. What happens if the condition \( X < \frac{1}{2} \) changes to another value?
5. How does the conditional expectation change if the interval is different?

**Tip:** When calculating conditional expectations, always redefine the probability density function according to the given condition.

Let X be uniform over (0, 1). Find E[X|X < 1/2].

Find the conditional expectation E[X|X < 1/2] where X follows a uniform distribution over (0, 1). This solution uses concepts from probability theory and uniform distributions, perfect for undergraduate students studying conditional expectations.

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