Download Discrete Probability Distribution

Chapter 5
Discrete Probability Distribution
5.1 Random Variables
Discrete
discrete values only
Continuous any values on a interval
Random variables is a function which assign a real number to each element of a sample space.
One sample space may be corresponding to more than one random variables.
e.g. Consider throwing 3 coins
random variable
“no. of T shown”
0 

1  


2  

3 
random variable
sample space
“no. of heads shown”
HHH  3
HHT 
HTH  2
THH 
HTT 
THT  1
TTH 
TTT  0
5.2 Probability Distributions
It can be shown by
1. a table
2. a line graph(bar chart)
3. a probability histogram - forget it
4. a mathematical formula(a function) - pdf
Probability Function(pdf)
A pdf must satisfy
1. 0  f(x)  for all x  A
 f (x)
2.
 1
P(X= x) = f(x)
A is the set of all possible values
xA
5.3 Expected Value (Expectation)
e.g. Consider throwing a die
a frequency distribution
of throwing 120 times
x
f
1
20
2
19
3
20
4
21
5
21
6
19

=
a probability distribution
x
prob.
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
1
fi xi
N
expected value =  Pixi
f
=  ( i ) xi
N
 = E(X) =  f(x)x
E(aX+b) = a E(X) + b
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5.4 Variance & Standard Deviation
recall the formula for frequency distribution
1
2 =
fi (xi - )2
N
f
=  ( i ) (xi - )2
N
variance of a discrete random variable X
Var(X) = 2 = f(x) (x - )2
= E [ (X - )2 ]
Var (a)
=0
Var(aX)
= a2 Var(x)
aX
= a X
Var(aX + b) = a2 Var(X)
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other formula
1
2 =
fi xi - 2
N
Var(X) = 2 = f(x)x2 - 2
= E(X2) - 2