Probability
and
Random Variables
Chapter 5
Multiple Random Variables
Note
S
Y
(X(si), Y(si))
s1
R2-plane
y
s2 ……. si
sj
x
P[X=x, Y=y]
= PX,Y(x,y)
P[si]
Note) * range of 𝑋, 𝑌 : 𝑆 , =
∗
𝑃 , 𝑥, 𝑦 = 1
∈
∈
𝑥, 𝑦
𝑃 , 𝑥, 𝑦 > 0 }
X
Note) * 𝐵 = 𝑋 = 𝑥 ⟺ 𝐵 =
𝑋 = 𝑥, 𝑌 = 𝑦
𝑦∈𝑆 }
∗ 𝑃 𝑥 =𝑃 𝐵 =
𝑃 , 𝑥. 𝑦 =
𝑃 , (𝑥, 𝑦)
,
∈
∈
Y
𝐵 = {𝑋 = 𝑥}
x
X
Note) * 𝑓 , 𝑥, 𝑦 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑎𝑟𝑒𝑎
* for small enough 𝑑𝑥 𝑎𝑛𝑑 𝑑𝑦,
𝑃 𝑥 < 𝑋 ≤ 𝑥 + 𝑑𝑥, 𝑦 < 𝑌 ≤ 𝑦 + 𝑑𝑦 = 𝑓 , 𝑥, 𝑦 𝑑𝑥𝑑𝑦
Note)
𝑥−𝜇
1−𝜌 ,
+ 𝑥−𝜇
=−
2𝜎
⋯
=−
𝜌 , −
𝑥−𝜇
2𝜎
1−𝜌 ,
𝜎
𝑥−𝜇 )
𝜎
2𝜎 (1 − 𝜌 , )
(𝑦 − 𝜇 − 𝜌 ,
−
2𝜌 , 𝜎 𝑥 − 𝜇
𝜎
𝑦−𝜇
+
𝜎
𝜎
𝑦−𝜇
Note)
𝑥−𝜇
=−
2𝜎
−
𝜎
(𝑦 − 𝜇 − 𝜌 , 𝜎
𝐿𝑒𝑡 𝜇 𝑥 ≜ 𝜇 + 𝜌 ,
𝑥−𝜇 )
2𝜎 (1 − 𝜌 , )
𝑥−𝜇
𝑎𝑛𝑑 𝜎 = 𝜎
1−𝜌 ,
.