Title Joint Distribution.pdf ✅

Joint Distribution When you go to a concert hall, sometimes you may want to see a solo violin concert, but other times you may want to see a symphony. Symphonies are appealing because many instruments are playing together. Random variables are similar. While single random variables are useful for modeling simple events, we use multiple random variables to describe complex events. The multiple random variables can be either independent or correlated. When many random variables are present in the problem, we enter the subject of joint distribution. Joint distributions are high-dimensional PMFs or CDFs. fX(x) | {z } one variable =⇒ fX1,X2 (x1, x2) | {z } two variables =⇒ · · · =⇒ fX1,X2,··· ,Xn (x1, x2, · · · ,xn) | {z } n variables 2.1 Joint PMF Definition 2.1.1: Joint PMF The function fXY : RX × RY −→ [0,1] is a joint probability distribution or probability mass function of the discrete random variables X and Y if • fXY (x,y) ≥ 0 ∀ (x,y). • P x P y fXY (x,y) = 1. • P(X = x, Y = y) = fXY (x,y). For any region A in the x − y plane, P((X, Y ) ∈ A) = P x∈A P y∈A fXY (x,y). 1