Download PROBABILITY DISTRIBUTION

PROBABILITY DISTRIBUTION
(distribusi peluang)
BUDIYONO
2011
RANDOM VARIABLES
(VARIABEL RANDOM)
 Suppose that to each point of sample space
we assign a real number
 We then have a function defined on the
sample space
 This function is called a random variable or
random function
 It is usually denoted by a capital letter such
as X or Y
RANDOM VARIABLES
(VARIABEL RANDOM)
S = {AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG}
The set of value of the above random variable is {0, 1, 2, 3}
A random variable which takes on a finite or countably infinite
number of values is called a discrete random variable
A random variable which takes on noncountably infinite
number of values ia called continous random variable
PROBABILITY FUNCTION
(fungsi peluang)
It is called probability function or probability distribution
Let X is a discrete random variable and suppose that it
values are x1, x2, x3, ..., arranged in increasing order of
magnitude
It assumed that the values have probabilities given by
P(X = xk) = f(xk), k = 1, 2, 3, ... abbreviated by
P(X=x) = f(x)
PROBABILITY FUNCTION
(fungsi peluang)
X
f
•
•
•
•
0.125
R
0.375
0.375
0.125
A function f(x) = P(X = x) is
called probability function of
a random variable X if:
random
variable
probability
function
1. f(x) ≥ 0 for every x in its
domain
2. ∑ f(x) = 1
Can it be a probability function?
 On a sample space A = {a, b, c, d}, it is defined the
function:
a. f(a) = 0.5;
f(b) = 0.3;
b. g(a) = 0.5; g(b) = 0.25;
c. h(a) = 0.5; h(b) = 0.25;
d. k(a) = 0.5; k(b) = 0.25;
f(c) = 0.3;
g(c) = 0.25;
h(c) = 0.125;
k(c) = 0.25;
f(d) = 0.1
g(d) = 0.5
h(d) = 0.125
k(d) = 0
 Solution:
a.
b.
c.
d.
f(x) is not a probability function, since f(a)+f(b)+f(c)+f(d)  1.
g(x) is not a probability function, since g(c)  0.
h(x) is a probability function.
k(x) is a probability function.
DENSITY FUNCTION
(fungsi densitas)
It is called probability density or density function
A real values f(x) is called density function if:
1. f(x) ≥ 0 for every x in its domain
2.

 f ( x )dx  1

It is defined that:
P(a<X<b) = P(a<X≤b) = P(a≤X<b) = P(a≤X≤b) =
b
 f ( x )dx
a
Can it be a density function?
a.
No, it is not. Since f(x) may be
negative
b.
No, it is not. Since the area is not 1
c.
Yes, it is. If the area is 1
area = 1
d.
Yes, it is. If the area is 1
area = 1
area = 1
(2,0)
Solution:
(2,0)
Solution:
Solution:
Solution:
Distribution Function for
Discrete Random Variable
Solution
Distribution Function for
Continuous Random Variable
Example
Solution:
MATHEMATICAL EXPECTATION
(nilai harapan)
Solution:
MATHEMATICAL EXPECTATION
(nilai harapan)
Solution:
The Mean and Variance of a
Random Variable
Solution:
So, we have:
Solution:
So, we have: