AP STATISTICS 7.3 Probability Distributions for Continuous Random Variables
7.3 Objectives: Ø Understand the definition and properties of continuous random variables Ø Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms
7.3 Objectives: Ø Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables. Ø Be able to calculate probabilities such as P(x < a), P(x a), and P(a x b)
Let X = the weight (in pounds) of a full-term newborn child (reported to the nearest pound x 4 5 6 7 8 9 p(x) 0.26 0.31 0.21 0.13 0.06 0.03
Consider the random variable: x = the weight (in pounds) of a full-term newborn child What type of variable is this?
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 displays the distribution of weights. The area of the rectangle centered over 7 pounds represents the probability 6.5 < x < 7.5
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 displays the distribution of weights. What is the sum of the areas of all the rectangles?
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 displays the distribution of weights. Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.
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 displays Notice the distribution that the rectangles of weights. are narrower and the histogram begins to Now suppose have that a smoother weight is appearance. reported to the nearest 0.1 pound. This would be the probability histogram.
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 displays If weight the distribution measured of weights. with greater and greater accuracy, the histogram Now suppose approaches that weight a smooth is reported curve. to the nearest 0.1 pound. This would be the probability histogram.
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 displays the distribution of weights. This is an example of a density curve. Now suppose that weight is reported to the nearest The 0.1 pound. shaded This area would represents be the the probability histogram. probability 6 < x < 8.
Probability Distributions for Continuous Variables Is specified by a curve called a density curve. The function that describes this curve is denoted by f(x) and is called the density function. The probability of observing a value in a particular interval is the area under the curve and above the given interval.
Properties of continuous probability distributions 1. f(x) > 0 (the curve cannot dip below the horizontal axis) 2. 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 f ( x ) 2(1 x ) = 0 The density curve is shown in the figure: 0 x 1 otherwise 2 Density 1 1 Tons
Gravel problem continued... What is the probability that at most ½ ton of gravel is sold during a randomly selected week? P(x < ½) = 1 ½(0.5)(1) =.75 2 Density The probability would be the shaded area under the curve and above the interval from 0 to 0.5. 1 1 Tons
P(x = ½) = The probability would be the area under the curve and above 0.5. 2 Density How do we find the area of a line segment? Since a line segment has NO area, then the probability that exactly ½ ton is sold equals 0. 1 1 Tons
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 = 1 ½(0.5)(1) =.75 2 1 Tons 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: 0.5 Density 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 0.5 Density 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 When the density is constant over an interval (resulting in a horizontal density curve), the probability distribution is called a uniform distribution. 0.5 Density 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.
The probability that a continuous random variable x lies between a lower limit a and an upper limit b is P(a < x < b) = (cumulative area to the left of b) (cumulative area to the left of a) P(a < x < b) = P(x < b) P(x < a)
Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is as pictured (a uniform distribution): a. What is the probability that x is less than 10 minutes? more than 15 minutes?
a. What is the probability that x is less than 10 minutes? more than 15 minutes?
b. What is the probability that X is between 7 and 12 minutes?
c. Find the value c for which P(x < c) =.9.
The graph gives the distribution of the yearly amount of rainfall in Rainy City: In a randomly selected year, (a) What is the probability that Rainy City got more than eight inches of rain? P(Rainy City got more than 8 inches of rain) = P(X > 8) = 1 P(X < 8) = 1 [0.44 + 0.30 + 0.15 + 0.06] = 1 0.95 = 0.05
The graph gives the distribution of the yearly amount of rainfall in Rainy City: In a randomly selected year, (b) What is the probability that Rainy City got between two and six inches of rain? P(Rainy City got between 2 and 6 inches of rain) = P(2 < X < 6) = 0.30 + 0.15 = 0.45
The graph gives the distribution of the yearly amount of rainfall in Rainy City: In a randomly selected year, (c) What is the probability that Rainy City got exactly two inches of rain? P(Rainy City got exactly 2 inches of rain) = 0
The graph gives the distribution of the yearly amount of rainfall in Rainy City: In a randomly selected year, (d) What is the probability that Rainy City got at most six inches of rain? P(Rainy City got at most 6 inches of rain) = P(X 6) = 0.44 + 0.30 + 0.15 = 0.89
7.3 Objectives: ü Understand the definition and properties of continuous random variables ü Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms
7.3 Objectives: ü Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables. ü Be able to calculate probabilities such as P(x < a), P(x a), and P(a x b)
For Tonight: HW: Read 7.3: Probability Distributions for Continuous Random Variables P414: 7.20, 7.21, 7.22, 7.23, 7.24, 7.26