Probability

1. Submitted To:
Maam Wajiha Nasir
Submitted By:
Muhammad Irfan Hussain
Roll # 39
Bs 3rd
Semester(Regular)

2. Binomial probability distribution:
Hyper geometric probability distribution
Poisson distribution
Poisson process
Negative binomial distribution
𝒇( 𝒙) = 𝒑( 𝑿 = 𝒙) = (
𝒏
𝒙
) 𝒑 𝒙
𝒒 𝒏−𝒙
, 𝒙
= 𝟎, 𝟏, 𝟐, … … … . . , 𝒏,
H( 𝒙; 𝑵, 𝒏, 𝒌) =
( 𝒌
𝒙)( 𝑵−𝒌
𝒏−𝒙)
( 𝑵
𝒏
)
, for x such that 0≤ 𝒙 ≥ 𝒏
𝐥𝐢𝐦𝒏→𝒙
𝒑→𝟎
𝒃( 𝒙; 𝒏, 𝒑) =
𝝁 𝒙
𝒆−𝝁
𝒙!
, 𝒙 = 𝟎, 𝟏, 𝟐 … … … ,∞
P( 𝒙; 𝞴𝒕) =
𝒆−𝞴𝒕( 𝞴𝒕) 𝒙
𝒙!
P(X=x)= ( 𝒙 − 𝟏
𝒌 − 𝟏
) 𝒑 𝒌
𝒒 𝒙−𝒌
, 𝒘𝒉𝒆𝒓𝒆 𝒙 = 𝒌 + 𝟏, 𝒌 + 𝟐, … … …

3. Geometric distribution
Multinomial distribution
Uniform distribution
Exponential distribution
𝒑( 𝑿 = 𝒙) = 𝒒 𝒙−𝟏
𝒑 , 𝒙=𝟏, 𝟐, 𝟑, … … … … . ∞
𝒑( 𝑿 𝟏=𝒙 𝟏,
𝑿 𝟐=𝒙 𝟐
… … … . 𝑿 𝑲=𝒙 𝒌
) =
𝒏!
𝒙 𝟏!, 𝒙 𝟐!, 𝒙 𝟑!, … … 𝒙 𝒌!
( 𝒙) = ∫
𝟏
𝑩 − 𝑨
𝑩
𝑨
, 𝑨 ≤ 𝑿 ≥ 𝑩,
𝒇( 𝒙) =
𝟏
𝜷
𝒆
−
𝒙
𝜷 , 𝟎 ≤ 𝒙 ≥ ∞

4. Normal distribution
Prepared by MUHAMMAD IRFAN HUSSAIN
Bs 3rd Semester 039 (Regular)
UNIVERSITY OF SARGODHA
 𝒇( 𝒙; 𝝁, 𝝈 𝟐) =
𝟏
𝝈√ 𝟐𝝅
𝒆
−
𝟏
𝟐𝝈 𝟐
( 𝒙−𝝁) 𝟐
− ∞ ≤ 𝒙 ≥ ∞
 𝒇( 𝒛; 𝒐, 𝟏) =
𝟏
√ 𝟐𝝅
𝒆
−
𝟏
𝟐𝝈 𝟐 𝒛 𝟐
𝟎 ≤ 𝒛 ≥ 𝟏
 𝒛 =
𝑿−𝝁
𝝈