L06.4 Conditional PMFs & Expectations Given an Event

TL;DR

Conditional probability models involve revising probabilities based on new information, including conditional PMFs and expectations.

Transcript

We now move to a new topic-- conditioning. Every probabilistic concept or probabilistic fact has a conditional counterpart. As we have seen before, we can start with a probabilistic model and some initial probabilities. But then if we are told that the certain event has occurred, we can revise our model and consider conditional probabilities that t... Read More

Key Insights

• 👶 Conditioning involves revising probabilities based on new information.
• 😒 Conditional PMFs are like ordinary PMFs, but they use conditional probabilities.
• ❓ The conditional mean and variance of a random variable are derived using conditional probabilities.
• 📏 Conditional probability models follow the same rules and theorems as ordinary probability models.
• 🥺 Conditional probability models reduce uncertainty and can lead to smaller variances.
• ❓ Conditional probability models involve using conditional probabilities throughout calculations.
• 🛩️ The variance in a conditional model can be smaller than the variance in the original model.

Questions & Answers

Q: What is a conditional probability model?

A conditional probability model involves revising probabilities based on new information. It uses conditional PMFs to represent the revised probabilities of events.

Q: How does a conditional PMF differ from an ordinary PMF?

A conditional PMF is similar to an ordinary PMF but applies to a new or revised conditional model. The entries in a conditional PMF still sum to 1, but the probabilities are conditional probabilities.

Q: How is the conditional mean of a random variable calculated?

The conditional mean of a random variable is calculated using the conditional probabilities. It is defined the same way as in the original case, but the calculation involves the conditional PMF.

Q: How does conditional probability affect the variance of a random variable?

In a conditional probability model, the variance of a random variable can change. If there is less uncertainty in the conditional model, the variance will be smaller compared to the original model.

Summary & Key Takeaways

• Conditional probability models involve revising probabilities based on new information.

• In a conditional probability model, the PMF is changed to a conditional PMF.

• The conditional mean and variance of a random variable are calculated using conditional probabilities.