Download Discrete random variables

Probability Refresher
COMP5416
Advanced Network Technologies
Discrete random variables
Probability
• Events can take only discrete values, e.g. integer
values
• Probabilities sum to 1.0
1.0   pk
Mean   kpk
Variance   k  mean 2 pk
1 2 3 4 5 6
value
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Simulation - 2
Example – Roll a die
• Discrete values 1,2,3,4,5,6
• Each with probability 1/6
Mean 
6
 1  1  2  3  4  5  6 21
k

 3.5
  6  
6
6
k 1
Variance 
6
2 1 
 k  3.5
k 1
   10.20833
6
Standard Deviation  variance  3.195048
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• Discrete values
• 1 with probability p,
• 0 with probability 1-p
Probability
Example – Bernoulli random variable
p
1-p
Mean  p
Variance  (1  p)  p
0
1
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value
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Example – Poisson random variable
k

pk  e  
Mean  
k!
Variance  
Probability
• Discrete values 0,1,2,...,infinity
• Value k has probability pk, with
0 1 2 3 4 5 6
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Simulation - 5
Continuous random variables
• Events can take real values on arbitrary range
• Probabilities integrate to 1.0
2
1.0   p( x)dx
1.8
1.6
Mean   xp( x)dx
1.4
1.2
Variance   x  mean  p( x)dx
1
0.8
0.6
2
0.4
0.2
3.
8
3.
6
3.
4
3
3.
2
2.
8
2.
6
2.
4
2
2.
2
1.
8
1.
6
1.
4
1
1.
2
0.
8
0.
6
0.
4
0
0.
2
0
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Continuous random variables
PrX  x
• Distribution Function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
3.
8
3.
6
3.
4
3.
2
3
2.
8
2.
6
2.
4
2.
2
2
1.
8
1.
6
1.
4
1.
2
1
0.
8
0.
6
0.
4
0.
2
0
0
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Simulation - 7
Example – negative exponential
random variable
Probability density function,
negative exponential, m=2
2
1.8
• Parameter m
1.4
1.2
 mx
1
0.8
0.6
 mx
0.2
m
3.
8
3.
6
3.
4
3
3.
2
2.
8
2.
6
2.
4
2
2.
2
1.
8
1.
6
1.
4
1
1.
2
0.
8
0.
6
Probability distribution function,
negative exponential, m=2
0.9
0.8
2
0.6
0.5
0.4
0.3
0.2
0.1
3.
8
3.
6
3.
4
3.
2
3
2.
8
2.
6
2.
4
2.
2
2
1.
8
1.
6
1.
4
1
1.
2
m
0.
8
0
0.
6
1
0.
4
Standard Deviation 
0.7
0
1
Variance   
m
1
0.
4
0
0
0.
2
PrX  x  1  e
1
Mean 
0.4
0.
2
p ( x)  me
1.6
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Example – Gaussian random variable
Gaussian density function,
m=0, s=1
0.45
0.4
• Parameters m, s
0.35
0.3
0.25
0.2
2
5
8
1
4
7
1.
1.
2.
2.
2.
3
1.
8
2.
1
2.
4
2.
7
3
9
1.
5
6
0.
1.
3
0.
0
0.
.6
.9
.2
.5
.8
.1
.4
.7
.3
-0
-0
-0
-1
-1
-1
-2
-2
Gaussian distribution function,
m=0, s=1
1
0.9
Variance  s
Standard Deviation  s
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1.
2
0.
9
0.
6
0.
3
0
0
-3
-2
.7
-2
.4
-2
.1
-1
.8
-1
.5
-1
.2
-0
.9
-0
.6
-0
.3
2
0
-3
2s 2
0.1
0.05
-2
1
p ( x) 
e
s 2
Mean  m

x  m 2

0.15
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Simulation - 9