Probability and Statisitcs

Transcription

1 Probability and Statistics Random Variables De La Salle University Francis Joseph Campena, Ph.D. January 25, 2017 Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

2 Outline Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

3 Definition Definition A random variable is a function that associates a real number to each element of the sample space of an experiment. Example In an experiment of tossing a fair coin twice, define the random variable X to be the number of heads in an outcome. The possible values of X are {0, 1, 2}. In an experiment of rolling a pair of dice, define the random variable Y to be the sum of the numbers on the top face of each dice (or the total number of dots on the top face of each dice). The set of all possible values of Y is {2, 3,..., 12}. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

4 Types of R.V. If a sample space contains a countable number of sample points, then it is called a discrete sample space. If a sample space contains an infinite number of sample points equal to the number of points on a line segment, then it is called a continuous sample space. A random variable is called a discrete random variable if its set of possible outcomes is countable. A random variable is called a continuous random variable if it can assume any value in some interval or intervals of real numbers. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

5 Probability Distributions Definition The set of ordered pairs (x, f (x)) is a probability function, probability mass function or probability distribution of a discrete random varialbe X if for each of the possible outcome x, 1. f (x) x f (x) = P(X = x) = f (x). Consider the experiment of tossing coin twice and the random variable X defined as the number of heads in the outcome. The probability distribution of X is X = x P(X = x) = f (x) Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

6 Properties of PMF The following are some important concepts in relation to the probability distribution of a discrete random variable X. The values x 1, x 2,..., x k of a discrete random variable for which is probability mass function is positive are called mass points. The function f (x) is usually called the probability mass function. To find the probability that the discrete random variable X will have a value between a to b, we get the sum f (x i ). a x i b Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

7 Example 1 A box contains three red marbles and four green marbles. Jay picks two marbles at random from this box. Define X to be the number of red marbles that jay picked. Construct a probability distribution of X. 2 A shipment of 8 microcomputers 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 that the school might have purchased. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

8 Definition The function f (x) is a probability density function for the continuous random variable X, defined over the set of real numbers, if the following are satisfied: 1. f (x) 0 for all x R. 2. f (x)dx = 1 3. P(a x b) = b a f (x)dx. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

9 Example Let X be a continuous random variable with probability density function { x 1 if 1.5 x 2.5 f (x) = 0 otherwise Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

10 Properties of PDF The following are some important concepts in relation to the probability distribution of a continuous random variable X. The set of values of a continuous random variable X for which the value of f (x) is positive is called its support. The function f (x) is usually called the probability density function. To find the probability that the continuous random variable X will have a value between a to b, we get the value of the integral b a f (x)dx. The probability that a continuous random variable will take on a particular value x is practically zero. That is, P(X = x) = 0. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

11 Mean and Varaince of a Discrete R.V. Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

12 Figure: Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

13 EXAMPLE Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

14 EXAMPLE Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

15 Fair Games Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

16 Fair Games Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17

17 EXAMPLES Find the value of the constant c such that the following is a pmf of a discrete random variable. Determine its mean and variance Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, / 17