3.1.3 Marginal Distributions Flashcards
(12 cards)
What is a marginal distribution?
It is the probability distribution of a subset of random variables derived from a joint distribution.
How do you find the marginal distribution of X from the joint distribution of X and Y (discrete)?
Sum the joint PMF over all values of Y:
P_X(x) = Σ_y P(X = x, Y = y)
How do you find the marginal distribution of Y from the joint distribution of X and Y (discrete)?
Sum the joint PMF over all values of X:
P_Y(y) = Σ_x P(X = x, Y = y)
When summing from a joint table, how do you find the marginal PMF of X?
Sum each column — since columns correspond to different values of Y.
When summing from a joint table, how do you find the marginal PMF of Y?
Sum each row — since rows correspond to different values of X.
How do you verify your marginal probabilities are correct?
Add them all up — the total should be 1.
In a probability table, what are you calculating when you sum across rows or columns?
A marginal distribution — rows for Y, columns for X.
What’s the difference between marginal probability and marginal distribution?
Marginal distribution gives all probabilities; marginal probability refers to a specific value (e.g. P(X = 2)).
If asked to calculate P(X = x), do you always need to write the full marginal distribution of X?
No — you can directly sum the relevant joint probabilities for that specific x.
When must you use the joint distribution instead of marginal?
When calculating a probability involving both variables, like P(X = x, Y = y).
When can you use either the marginal or the joint distribution?
When the probability only involves one variable (e.g. P(X = 2)).
