PROBABILITY DISTRIBUTION

BUDIYONO

2011

(distribusi peluang)

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

R• 0.125

• 0.375

• 0.375

• 0.125f

random variable

probability function

A function f(x) = P(X = x) is called probability function of a random variable X if:

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; f(c) = 0.3; f(d) = 0.1b. g(a) = 0.5; g(b) = 0.25; g(c) = 0.25; g(d) = 0.5c. h(a) = 0.5; h(b) = 0.25; h(c) = 0.125; h(d) = 0.125d. k(a) = 0.5; k(b) = 0.25; k(c) = 0.25; k(d) = 0

Solution:a. f(x) is not a probability function, since f(a)+f(b)+f(c)+f(d) 1.

b. g(x) is not a probability function, since g(c) 0.c. h(x) is a probability function.d. k(x) is a probability function.

DENSITY FUNCTION(fungsi densitas)

1dx)x(f

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.

It is defined that:

P(a<X<b) = P(a<X≤b) = P(a≤X<b) = P(a≤X≤b) =

b

adx)x(f

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 1area = 1

d.Yes, it is. If the area is 1

area = 1

Solution:(2,0)

area = 1

Solution:(2,0)

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: