Jointly continuous PDF In order for a function f(x,y)to be a joint density it must satisfy f(x,y)≥0 fir(x.y)dxdy =1 6
Jointly continuous PDF • In order for a function 𝑓(𝑥, 𝑦) to be a joint density it must satisfy 𝑓(𝑥, 𝑦) ≥ 0 𝑓𝑋,𝑌 𝑥, 𝑦 𝑑𝑥 ∞ −∞ 𝑑𝑦 ∞ −∞ = 1 6
Jointly continuous PDF Just as with one random variable,the joint density function contains all the information about the underlying probability measure if we only look at the random variables X and Y. In particular,we can compute the probability of any event defined in terms of X and Y just using f(x,y). 7
Jointly continuous PDF • Just as with one random variable, the joint density function contains all the information about the underlying probability measure if we only look at the random variables 𝑋 and 𝑌 . • In particular, we can compute the probability of any event defined in terms of 𝑋 and 𝑌 just using 𝑓(𝑥, 𝑦). 7
Jointly continuous PDF Here are some events defined in terms of X and Y: ·X≤Y ·{X2+Y2≤1} ·{1≤X≤4，Y≥0 They can all be written in the form {(X,Y)E B}for some subset B of R2. 8
Jointly continuous PDF Here are some events defined in terms of 𝑋 and 𝑌 : • 𝑋 ≤ 𝑌 • {𝑋 2+ 𝑌 2 ≤ 1} • {1 ≤ 𝑋 ≤ 4, 𝑌 ≥ 0} They can all be written in the form {(𝑋, 𝑌) ∈ 𝐵} for some subset 𝐵 of 𝑅 2 . 8