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Chapter 3: Problem 14

Let \(X\) be uniform over \((0,1)\). Find \(E\left[X \mid X<\frac{1}{2}\right]\).

Short Answer

Expert verified
The conditional expectation \(E\left[X \mid X<\frac{1}{2}\right]\) is \(\frac{1}{4}\).

Step by step solution

01

Identify the pdf of \(X\)

Since \(X\) is uniform over \((0,1)\), the pdf will be constant over this interval. The following is pdf of \(X\): \( p(x) = 1, \) for \(0 \leq x < 1 \), otherwise, \( p(x) = 0 \).
02

Determine the new range of \(X\) given the condition \(X < \frac{1}{2}\)

Given the condition \(X<\frac{1}{2}\), the new range of \(X\) will be from 0 to \(\frac{1}{2}\), inclusive of 0, but exclusive of \(\frac{1}{2}\). The pdf of \(X\) remains the same, but over this new range.
03

Find the conditional expectation

Now, we will calculate the conditional expectation \(E\left[X \mid X<\frac{1}{2}\right]\) which is the mean of the variable over the new range (0 to \(\frac{1}{2}\)). The formula for finding the mean in a uniform distribution over the range (a,b) is given by \(\frac{a+b}{2}\). Substituting in our values we get: \[E\left[X \mid X<\frac{1}{2}\right] = \frac{0+ \frac{1}{2}}{2} = \frac{1}{4}\]. So, the expected value of \( X \) given that \( X < \frac{1}{2} \) is \( \frac{1}{4} \).

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