Probability and
Stochastic Processes
Vector Random Variable
By Dr Ayyem Pillai V, Professor, ECE,
GRIET, Hyderabad
Vector Random Variable

Consider a set of random variables 𝑋1 ,
𝑋2 ….., 𝑋𝑁 . Let sample values be written
as 𝑥1 𝑥2……, 𝑥𝑁.
The random variable and random sample
value vectors are written as follows.
𝑋1
𝑋
X= 2
⋮
𝑋𝑁
𝑥1
𝑥
and x= 2
⋮
𝑥𝑁
Vector Random Variable
Joint event of X is notated as
X≤ 𝒙.
X≤ 𝒙=𝑋1 ≤ 𝑥1 , 𝑋2 ≤ 𝑥2 ……, 𝑋𝑁 ≤ 𝑥𝑁
Joint CDF of joint event X≤ 𝑥 is
notated as P(X≤ 𝑥 ).
P(X≤ 𝒙) = 𝐹𝑿 (𝒙) = P(𝑋1 ≤ 𝑥1 , 𝑋2 ≤
𝑥2 ……, 𝑋𝑁 ≤ 𝑥𝑁 )

Joint PDF is defined as follows.
𝑓𝐗 (x)=𝑓𝑋1,𝑋2,…..,𝑋𝑁 (𝑥1 𝑥2 ……, 𝑥𝑁 )

𝜕𝑁 𝐹𝑋1 ,𝑋2 ,…..,𝑋𝑁 (𝑥1 𝑥2 ……, 𝑥𝑁 )
𝜕𝑁 𝐹𝑿 (𝒙)
=
=
𝜕𝑥1 𝜕𝑥2 …..𝜕𝑥𝑁
𝜕𝑥1 𝜕𝑥2 …..𝜕𝑥𝑁
𝐹𝑋1,𝑋2,…..,𝑋𝑁 (𝑥1 𝑥2 ……,
∞ ∞
∞
𝑥𝑁 )= −∞ −∞⋯ −∞ 𝑓𝑋1,𝑋2,…..,𝑋𝑁 (𝑥1 𝑥2 ……,
𝑥𝑁 ) 𝑑 𝑥𝑁 𝑑𝑥𝑁−1 ⋯ 𝑑𝑥1

𝐹𝑋,𝑌 (𝑥,y)=P(𝑋 ≤ 𝑥 , Y ≤ 𝑦)
𝜕2 𝐹𝑋,𝑌 (x,y)
𝑓𝑋,𝑌 (𝑥, 𝑦 )=
𝜕𝑥 𝜕𝑦
𝐹𝑋,𝑌 (x,y)=
𝑥
𝑦
𝑓
−∞ −∞ 𝑋,𝑌
(𝑢, 𝑣 )dv du