L09.3 Conditioning Example

TL;DR

The conditional probability density function (PDF) is a constant within a specified range and follows the shape of the unconditional PDF.

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 probability density function?

The conditional PDF is a constant within the specified interval of conditioning and 0 outside that interval. It retains the same shape as the unconditional PDF within the allowed range.

Q: How can we determine the height of the conditional PDF?

The height of the conditional PDF is determined by the length of the interval. The height is calculated as 2 divided by the difference between b and a.

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

The conditional expectation is the ordinary expectation of the conditional model. In this case, it is the midpoint of the range of the conditional PDF, which evaluates to 1/4 times a plus 3/4 times b.

Q: How do we calculate the expected value of X squared in the conditional model?

The expected value of X squared in the conditional model is calculated using the expected value rule. It involves multiplying the conditional PDF by x squared and integrating over the range 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.