Lecture 1.3: Conditional distribution of one random variable given another

Summary

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Takeaways

• 😀 Conditional PMF (Probability Mass Function) is defined for a random variable X given an event A, representing the probability of X under the condition of A occurring.
• 😀 The conditional PMF can differ from the original PMF, as conditioning may limit the range of possible values for the random variable.
• 😀 The formula for computing the conditional PMF is the probability of the intersection of events divided by the probability of the conditioning event.
• 😀 When defining the conditional PMF of a random variable Y given another random variable X, we utilize joint PMF to find the probabilities.
• 😀 The notation Y given X equals t indicates that Y is conditioned on the occurrence of X taking the value t.
• 😀 The relationship between joint PMF and conditional PMF can be expressed through a product rule, similar to the concept of conditional probability.
• 😀 Conditional distributions can be computed easily by dividing the relevant entries in the joint PMF by the marginal PMF of the conditioning variable.
• 😀 Each conditional PMF is itself a valid PMF, meaning that when summed over its range, it equals 1.
• 😀 The values of the conditional PMF change depending on the conditioning event, showcasing the dependence between random variables.
• 😀 Key identities in probability allow for the calculation of missing values when partial information about joint and conditional PMFs is provided.

Q & A

What is a conditional PMF?

A conditional PMF is the probability mass function of a random variable given that a certain event has occurred. It provides the probability of a random variable taking on specific values under the condition of an event.

How do you compute the conditional PMF of a random variable X given an event A?

The conditional PMF of X given A is computed using the formula P(X = t | A) = P(X = t ∩ A) / P(A), where P(X = t ∩ A) is the probability of the intersection of the two events.

What happens to the range of a random variable when conditioning on an event?

The range of the conditioned random variable can differ from the original random variable's range, as conditioning on an event A may restrict the values that the random variable can take.

What is the relationship between joint PMFs and conditional PMFs?

The relationship is defined by the product rule: the joint PMF P(X = t, Y = t') can be expressed as P(Y = t' | X = t) * P(X = t), allowing for the calculation of joint probabilities using conditional and marginal probabilities.

How can you derive the conditional PMF of Y given X = t?

To derive the conditional PMF of Y given X = t, use the formula P(Y = t' | X = t) = P(Y = t', X = t) / P(X = t), where the numerator is the joint PMF and the denominator is the marginal PMF of X.

What is an important identity relating to joint PMFs and conditional PMFs?

The identity states that P(X = t1, Y = t2) = P(Y = t2 | X = t1) * P(X = t1), which holds true for all random variables and is essential for calculations involving joint distributions.

Why is it important to check the validity of a PMF?

It is important to check that the PMF sums to 1 across all possible values of the random variable, as this ensures the probabilities are correctly defined and that they represent a valid probability distribution.

What is the procedure for calculating conditional PMFs from a joint PMF table?

To calculate a conditional PMF from a joint PMF table, identify the relevant row or column corresponding to the condition, divide the joint PMF values by the marginal PMF value associated with the condition.

Can the conditional PMF itself be considered a valid PMF?

Yes, the conditional PMF is a valid PMF on its own. When conditioned on a specific value of X, summing the probabilities across the valid range of Y will yield 1.

What role does conditioning play in understanding the relationship between random variables?

Conditioning allows for a clearer understanding of the relationship between random variables by defining the distribution of one variable given the known value of another, helping to uncover dependencies and interactions.