How to Find the Conditional PDF and Expectation

TL;DR

The conditional probability density function (PDF) is zero outside the interval from (a + B)/2 to B and is constant at 2/(B − a) within that range. The conditional expectation is the midpoint of the interval, calculated as a/4 + 3B/4, and the expected value of X squared can be found by integrating the conditional PDF multiplied by x squared over the appropriate range.

Transcript

Let us now look at an example. Consider a piecewise constant PDF of the form shown in this diagram. Suppose that we condition on the event that x lies between a plus b over 2, which is here, and b. So we're conditioning on x lying in this particular red interval. What is the conditional PDF? The conditional PDF is going to be 0 outside of the inter... Read More

Key Insights

• ❓ The conditional PDF is 0 outside the conditioned interval.
• 👻 The conditional PDF retains the shape of the unconditional PDF within the allowed range.
• ❓ The height of the conditional PDF is determined by the length of the interval.
• 🧡 The conditional expectation is the midpoint of the range of the conditional PDF.
• ☺️ The expected value of X squared in the conditional model can be calculated using the expected value rule.
• ✖️ The expected value rule involves integrating the conditional PDF multiplied by x squared over the nonzero range.
• ❓ The height of the conditional PDF is inversely proportional to the length of the conditioning interval.

Questions & Answers

Q: What is the conditional PDF when X lies between (a + B)/2 and B?

The conditional PDF is zero outside the interval from (a + B)/2 to B. Within that interval, it is constant with height 2/(B − a).

Q: Why is the conditional PDF zero outside the conditioning interval?

The conditioning information allows only values of X between (a + B)/2 and B. Therefore, the conditional PDF is zero in every range outside that interval.

Q: Why does the conditional PDF remain constant within the allowed interval?

Within the allowed range, the conditional PDF retains the same shape as the unconditional PDF. Because the unconditional PDF is constant there, the conditional PDF is also constant.

Q: What is the length of the conditioning interval?

The interval runs from (a + B)/2 to B. Its length is half the distance B − a, which equals (B − a)/2.

Q: How is the height 2/(B − a) determined?

The area under the conditional PDF must equal one. Since the allowed interval has length (B − a)/2, the constant height must be 2/(B − a).

Q: How is the conditional expectation found in this example?

The conditional expectation is the ordinary expectation applied to the conditional model. Since the conditional PDF is uniform, the expectation is the midpoint of its interval.

Q: What is the conditional expectation of X?

The midpoint is one-half times the left endpoint, (a + B)/2, plus one-half times the right endpoint, B. This evaluates to a/4 + 3B/4.

Q: How is the conditional expected value of X squared set up?

Apply the expected value rule by multiplying x squared by the conditional PDF, 2/(B − a), and integrating. The integral runs from (a + B)/2 to B, where the conditional PDF is nonzero.

Summary & Key Takeaways

• The conditional PDF is 0 outside the interval on which we are conditioning.

• Within the allowed range, the conditional PDF retains the same shape as the unconditional PDF.

• The height of the conditional PDF is determined by the length of the interval.