3.2.1 Conditional Distributions Flashcards
(14 cards)
What is a conditional distribution?
It’s the distribution of one random variable given a fixed value (or condition) on another.
What is the conditional distribution of X given Y = y (discrete)?
P(X = x | Y = y) = P(X = x ∩ Y = y) / P(Y = y)
When is it necessary to calculate both the numerator and denominator separately?
When the condition involves a range of values (e.g., Y ≥ 1) rather than a specific value.
How do you find the conditional distribution when the condition is like Y ≥ 1?
Sum P(X = x, Y = y) over the conditional domain (e.g., all y ≥ 1), then normalize the values.
What’s a good rule of thumb for conditional distributions?
Identify the conditional domain → sum joint PMFs over that domain → normalize.
What is a conditional expectation?
The expected value of a random variable given a condition on another variable.
What is the formula for conditional expectation E[X | Y = y]?
E[X | Y = y] = Σ_x x * P(X = x | Y = y)
When the condition is on a range of values (e.g. Y > x), how do you compute E[X | Y > x]?
Use: E[X | Y > x] = (Σ_x Σ_y x * P(X = x, Y = y)) / P(Y > x)
How do you find the variance of a conditional distribution?
Var(X | Y = y) = E[X² | Y = y] - (E[X | Y = y])²
In a joint distribution table, how do you compute P(X = x | Y = y)?
Divide the joint cell P(X = x, Y = y) by the marginal total for Y = y
What is a common mistake when working with conditional distributions?
Forgetting to normalize the conditional distribution — all probabilities must sum to 1.
What does it mean if P(X = x | Y = y) = P(X = x)?
X and Y are independent
How do you know if a conditional distribution is “still random”?
If the condition involves a range or inequality (e.g., Y ≥ 2), then X is still random — not fixed.
What’s the general form of the conditional PMF for P(X = x | event E)?
P(X = x | E) = P(X = x ∩ E) / P(E), where E is any event involving X and/or Y
