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 x 1, x 2, x 3,..., arranged in increasing order of magnitude It assumed that the values have probabilities given by P(X = x k ) = f(x k ), k = 1, 2, 3,... abbreviated by P(X=x) = f(x)

PROBABILITY FUNCTION (fungsi peluang) X R f 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) = 0.1 b. g(a) = 0.5; g(b) = 0.25; g(c) =  0.25; g(d) = 0.5 c. h(a) = 0.5; h(b) = 0.25; h(c) = 0.125; h(d) = d. 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) 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) =

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

Solution: (2,0) area = 1

Solution: (2,0)

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: