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Chapter 1, Introduction to Probability and Statistics (1) Information comes with uncertainty. For example whether a friend will call today the time that a train will arrive the time that a machine will fail …… In order to study these phenomena, we need probability and statistics (2) Probability and statistics is a study of uncertain events. It gives us a quantitative way to measure the uncertainty. (3) An example: take a survey from the students in the front row: record name (x1) 1 Laura 2 Annit 3 Karen 4 Joe 5 Rob 6 Josh 7 Germmy 8 Hugh 9 Matt 10 Jennifer 11 Jair 12 Mat sex (x2) F F F M M M M M M F M M hours of study per week (x3) 15 12 15 12 14 10 25 20 14 20 15 14 hours of exercise per week (x4) 7 0 5 6 4 10 10 8 0 5 4 4 (4) What is the average? the questions: average hours of study per week average hours of study per week of male students average hours of study per week of female students average hours of exercise per week ...... the measure of average: the mean: 1 n x xi n i1 the medium: sort the data in ascending order the datum at the (n+1)/2 position is the medium the mode 1-1 arrange the data by incremental intervals the position at which most data is concentrated is the mode in the above example x 3 = 15.5 (the mean hours of study per week) x 4 = 5.25 (the mean hours of exercise per week) also, for male student: x 3M = 15.5 x 4M = 5.75 for the female student x 3F = 15.5 x 4F = 4.25 note different data sets may have a same mean, for example: A = {1, 1, 1, 1} B = {2, 2, 0, 0} but, xA= xB=1 therefore, it is necessary to have another way to measure the uncertain events: variation. (5) What is the variation? the questions: the variation of hours of study per week the variation of hours of study per week of male students ...... the measure of variation the variance 1 n s2 xi x 2 n 1 i1 the standard deviation: s the range: r = max{xi} - min{xi} in the above example: sx3 = 4.19 sx4 = 3.25 sx3M = 4.78 sx4M = 3.45 sx3F = 3.32 sx4F = 2.98 1-2 observations: - each student is somewhat different - the statistics would be different (e.g., the mean and the variance of male and female students) - female students are more consistent than the male students (because of the variation is small). - note the above statistics are true ONLY for the 12 students in the front row. What about the students in the entire class? the students in the University? .... (6) Unanswered questions: the questions: - what is the hours of study per week of a students in the university? (prediction) - is there any correlation between the hours of study and the hours of exercise? (correlation) - ...... the answers can be found by the probability and statistical theory, which we will learn in the first part of this course. 1-3