Slide 1Motivation ~ Consider the following discrete random variable…Slide 3Motivation ~ Consider the following discrete random variable…Motivation ~ Consider the following discrete random variable…Slide 6Slide 7Slide 8Slide 9Slide 10Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Consider the following continuous random variable…Slide 28CHAPTER 4• 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc.• 4.2 - Continuous Models General distributions Classical: Normal, etc.X161616161616Motivation ~ Consider the following discrete random variable…2Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Probability Tablex f(x)1 1/62 1/63 1/64 1/65 1/66 1/61Probability Histogram“What is the probability of rolling a 4?”( 4)P X = =X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Total Area = 1P(X = x)Density(4)f =X1616161616163Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Probability Tablex f(x)1 1/62 1/63 1/64 1/65 1/66 1/61Probability Histogram“What is the probability of rolling a 4?”( 4)P X = =16X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Total Area = 1P(X = x)Motivation ~ Consider the following discrete random variable…Density(4)f =Motivation ~ Consider the following discrete random variable…4Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” P(X = x)x f(x)1 1/62 1/63 1/64 1/65 1/66 1/61X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(X  x)F(x)1/62/63/64/65/61Motivation ~ Consider the following discrete random variable…5Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” P(X = x)x f(x)1 1/62 1/63 1/64 1/65 1/66 1/61X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(X  x)F(x)1/62/63/64/65/61“staircase graph” from 0 to 1Time intervals = 0.5 secsTime intervals = 2.0 secsTime intervals = 1.0 secsTime intervals = 1.0 secsTime intervals = 5.0 secs“In the limit…”POPULATIONrandom variable XContinuous6Example: X = “reaction time”“Pain Threshold” Experiment:Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn.“Pain Threshold” Experiment:Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn.In principle, as # individuals in samples increase without bound, the class interval widths can be made arbitrarily small, i.e, the scale at which X is measured can be made arbitrarily fine, since it is continuous.SAMPLETotal Area = 1we obtain a density curvex7“In the limit…”xCumulative probability F(x) = P(X  x) = Area under density curve up to x f(x) no longer represents the probability P(X = x), as it did for discrete variables X. •f(x)  0•Area = 1f(x) = density function00xIn fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)However, F(x) increases continuously from 0 to 1.we can define “interval probabilities” of the form P(a  X  b), using F(x). we obtain a density curvef(x) no longer represents the probability P(X = x), as it did for discrete variables X. 8“In the limit…”Cumulative probability F(x) = P(X  x) = Area under density curve up to x •f(x)  0•Area = 1f(x) = density functionF(x) increases continuously from 0 to 1.ababHowever, we can define “interval probabilities” of the form P(a  X  b), using F(x). F(a)F(b)F(b)  F(a)we obtain a density curveIn fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)An “interval probability” P(a  X  b) can be calculated as the amount of area under the curve f(x) between a and b, or the difference P(X  b)  P(X  a), i.e., F(b)  F(a). (Ordinarily, finding the area under a general curve requires calculus techniques… unless the “curve” is a straight line, for instance. Examples to follow…)f(x) no longer represents the probability P(X = x), as it did for discrete variables X. 9“In the limit…”Cumulative probability F(x) = P(X  x) = Area under density curve up to x •f(x)  0•Area = 1f(x) = density functionababF(x) increases continuously from 0 to 1.F(a)F(b)F(b)  F(a)we obtain a density curveMoreover, and . ( ) ( ) .s m+�- �= -�2 2x f x dx( )m+�- �=�x f x dx10f(x) = density functionCumulative probability F(x) = P(X  x) = Area under density curve up to x Thus, in general, P(a  X  b) = = F(b)  F(a). ( )�bf x dxa“In the limit…”•f(x)  0•Area = 1( )+�- �=�1f x dxF(x) increases continuously from 0 to 1.Fundamental Theorem of Calculuswe obtain a density curveX161616161616Consider the following continuous random variable…11Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” “What is the probability of rolling a 4?”( 4)P X = =1 6Probability HistogramTotal Area = 1Probability Tablex f(x)1 1/62 1/63 1/64 1/65 1/66 1/61P(X = x)F(x)1/62/63/64/65/61Cumul ProbP(X  x)“staircase graph” from 0 to 1DensityX161616161616Consider the following continuous random variable…12Example: X = “Ages of children from 1 year old to 6 years old”“What is the probability of rolling a 4?”( 4)P X = =Further suppose that X is uniformly distributed over the interval [1, 6]. Probability HistogramTotal Area = 1Probability Tablex f(x)1 1/62 1/63 1/64 1/65 1/66 1/61P(X =