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Chapter 7 Lesson 7.3 Random Variables and Probability Distributions 7.3 Probability Distributions for Continuous Random Variables Probability Distributions for Continuous Random Variables Consider the random variable: x = the weight (in pounds) of a full-term newborn child Suppose that weight is reported to the nearest pound. The following probability histogram What type of variable is this? If weight is measured with greater What is the sum of the areas of all displays the distribution of weights. The area of the rectangle and greater accuracy, thecentered histogram the rectangles? Notice that the rectangles are The shaded area represents the over 7 pounds represents the This is an example approaches ahistogram smooth curve. Nownarrower suppose that and the weight is reported begins to to the probability 6 < x < 8. probability 6.5 appearance. < xof < 7.5 a density curve. have a smoother nearest 0.1 pound. This would be the probability histogram. Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The probability of observing a value in a particular interval is the area under the curve and above the given interval. • The total area under the density curve equals one. Let x denote the amount of gravel sold (in tons) during a randomly selected week at a particular sales facility. Suppose that the density curve has a height f(x) above the value x, where 2(1 x ) 0 x 1 f (x ) otherwise 0 The density curve is shown in the figure: Density 2 1 Tons 1 Gravel problem continued . . . What is the probability that at most ½ ton of gravel is sold during a randomly selected week? P(x < ½) = .75 Density 2 The probability would be the shaded area under the curve and above the interval from 0 to 0.5. 1 Tons 1 Gravel problem continued . . . What is the probability that exactly ½ ton of gravel is sold during a randomly selected week? P(x = ½) = 2 0 The probability would be the area Since a line segment has NO area, Density the curve that and above 0.5. thenunder the probability exactly ½ ton is sold equals 0. 1 Tons 1 Gravel problem continued . . . What is the probability that less than ½ ton of gravel is sold during a randomly selected week? P(x < ½) = P(x < ½) Density 2 1 = .75 Does theHmmm probability change . . . This is whether different the ½ is included or not? than discrete probability distributions where it does change the probability whether a value is included or Tons not! 1 Suppose x is a continuous random variable defined as the amount of time (in minutes) taken by a clerk to process a certain type of application form. Suppose x has a probability distribution with density function: .5 4 x 6 f (x ) 0 otherwise The following is the graph of f(x), the density curve: Density 0.5 4 5 6 Time (in minutes) Application Problem Continued . . . What is the probability that it takes more than 5.5 minutes to process the application form? P(x > 5.5) = .5(.5) = .25 Find the probability by calculating the area of the shaded region (base × height). Density 0.5 4 5 6 Time (in minutes) Other Density Curves Some density curves resemble the one below. Integral calculus is used to find the area under the these curves. Don’t worry – we will use tables (with the values already calculated). We can also use calculators or statistical software to find the area. Homework • Pg.320: #7.20-22, 25