Random Variables and Probability Distribution (1)

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Presentation transcript:

Random Variables and Probability Distribution (1) Definition: A random variable is a function that associates a real number with each element in the sample space. A random variable symbol should be capital letter such X, Y, … and its corresponding is small letter.

Random Variables and Probability Distribution (1) Example: Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. The possible outcomes and the value Y of the random variable Y where Y is the number of red balls are: Sample space “S”: [RR, RB, BR, BB] Number of reds “Y”: [2, 1, 1, 0]

Random Variables and Probability Distribution (1) Definition: If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space.

Random Variables and Probability Distribution (1) Definition: If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space.

Random Variables and Probability Distribution (1) Example: Classify the following random variables are discrete or continuous: X: The number of automobile accidents? It is discrete random variable. Y: The length of time to play 18 holes of Golf? It is continuous random variable. M: The mount of milk produced yearly by a particular cow?

Random Variables and Probability Distribution (1) Example: N: The number of eggs laid each month by a hen? It is discrete random variable. P: The number of building permits issued each monthly in a certain city? Q: The weight of grain produced per acre? It is continuous random variable.

Random Variables and Probability Distribution (1) Discrete Probability Distribution Definition : The set of ordered pairs (x, f(x)) is a probability function, probability mass function or probability distribution of the discrete random variable X if for each possible outcome x ;

Random Variables and Probability Distribution (1) As

Random Variables and Probability Distribution (1) Example (1): A shipment of 20 similar laptop computers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.

Random Variables and Probability Distribution (1) Solution: Let X be a random variable whose values x are the possible numbers of defective computers purchased by the school. Then x can only take the numbers 0, 1, 2 then:

Random Variables and Probability Distribution (1) 2.

Random Variables and Probability Distribution (1) Then the probability distribution is X 1 2 f(x) 68/95 51/190 3/190

Random Variables and Probability Distribution (1) Example2: Suppose X is the sum of up faces of 2 dies:

Random Variables and Probability Distribution (1) X: sum of up faces of 2 dies 1 2 3 4 5 6 7 8 9 10 11 12

Random Variables and Probability Distribution (1) X f(x) = P (X=x) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 9 10 3/36 11 12 1/36 The probability distribution function table is

Random Variables and Probability Distribution (1) Then