# Some Popular Probability Distributions for Discrete Random Variables ```Some Popular Probability Distributions for
Discrete Random Variables
1. Binomial Distribution
 Situation: Sampling from an infinite (or very large) population
 Information supplied:
p = probability of &quot;success&quot;
n = sample size
 Asked to compute: The probability of occurrence of x successes in a sample
of size n.
 If X ~ Bin(n, p), then:
n
(i) P(X = x) =   p x (1  p) n x
 x
(ii)   np
(iii)  2  npq
2. Hypergeometric Distribution
 Situation: Sampling from a finite (or small) population
 Information supplied:
N = population size
n = sample size
k = no. of successes in the population
 Asked to compute: The probability of occurrence of x successes in a sample
of size n.
 If X ~ Hyp(N, n, k), then:
 k  N  k 
 

x  n  x 

(i) P(X=x) =
N
 
n
nk
(ii)  
N
nk 
k  N  n 
(iii)  2  1  

N  N  N  1 
3. Poisson Distribution
 Situation: Event occurring within a fixed interval or fixed length of time
 Information supplied:
Interval length
 = average no. of times that an event happens within the given interval
 Asked to compute: The probability of an event occurring x times within the
given interval.
 If X ~ Poi(  ), then:
x e 
(i) P(X = x) =
x!
(ii)   
(iii)  2  
Some Popular Probability Distributions for
Continuous Random Variables
1. Exponential Distribution
 1  x
 e , x0
f ( x)   
otherwise
0,

1
 If X ~ exp 


 , then (i) E(X) =  , and (ii) var(X) =  2

2. Normal Distribution
1

 x   2
,   x  
2 2
 If X ~ N  , 2 , then: (i) E(X) =  , and (ii) var(X) =  2
X 
 If X ~ N  , 2 , then the random variable Z 
has a standard normal
f ( x) 


e
2 2



distribution, i.e. Z ~ N (0,1) .
3. Chi-squared Distribution
 used in the inference of variance
 for a sample of size n, we apply the Chi-squared  2
  n 1 degrees of freedom.
 
distribution with
4. T Distribution
 approximates the normal distribution when the sample size is small (n &lt; 30).
 used in the inference of mean from a small sample
 for a sample of size n, we apply the T distribution with   n 1 degrees of
freedom.
Sampling Theory
Suppose that a sample of size n is drawn from a population P with E(P) =  and
var(P) =  2 . Let the mean and variance of the sample be X n and S n2 respectively.


(i) If the sample size is small (n &lt; 30), and if P ~ N  , 2 , then
(ia) when  2 is known, then the sample mean X n has approximately a normal
 2 
2
.
distribution with mean  and variance
, i.e. X n ~ N   ,
n
n 

(ib) when  2 is unknown (or not given), then the transformed sample mean,
Xn 
T
, follows a t distribution with   n 1 degrees of freedom.
sn n


(ii) If the sample size is sufficiently large (n  30), then regardless of P's
 2 
 . Use s n2 in place of  2 if  2 is unknown.
distribution, X n ~ N   ,
n 

```