Muddy Points

Modified

November 3, 2025

Fall 2023

1. Joint vs marginal vs conditional: How are we calculating the probability?

If we start at a joint probability \(f_{X,Y}(x,y)\)…. we can look at a few probabilities:

  • Joint probability: \(P(a \leq X \leq b, c \leq Y \leq d)\)

    \[P(a \leq X \leq b, c \leq Y \leq d) = \displaystyle\int_{x=a}^{x=b}\displaystyle\int_{y=c}^{y=d} f_{X,Y}(x,y) dydx\]

  • Marginal probability: \(P(a \leq X \leq b)\)

    \[P(a \leq X \leq b) = \displaystyle\int_{x=a}^{x=b} f_{X}(x) dx\]

    OR

    \[P(a \leq X \leq b) = \displaystyle\int_{x=a}^{x=b}\displaystyle\int_{y=-\inf}^{y=\inf} f_{X,Y}(x,y) dydx\]

  • Conditional probability: \(P(a \leq X \leq b | Y = c)\)

    \[P(a \leq X \leq b | Y=c) = \displaystyle\int_{x=a}^{x=b} f_{X|Y}(x|y=c) dx\]

    You cannot calculate \(P(a \leq X \leq b | Y = c)\) by \(\dfrac{P(a \leq X \leq b, Y=c)}{P(Y = c)}\) because \(P(Y = c)\) is 0. Instead, we need to find \(f_{X|Y}(x|y=c)\) by \(\dfrac{f_{X,Y}(x,y=c)}{f_{Y}(y=c)}\) and THEN integrate over X.