Random Variables Lesson 5 5 Continuous Random Variables

Random Variables Lesson 5. 5 Continuous Random Variables Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Continuous Random Variables Learning Targets After this lesson, you should be able to: ü Show that the probability distribution of a continuous random variable is valid and use the distribution to calculate probabilities. ü Determine the relative locations of the mean and median of a continuous random variable from the shape of its probability distribution. ü Draw a normal probability distribution with a given mean and standard deviation. Statistics and Probability with Applications, 3 rd Edition 2

Continuous Random Variables Below is a relative frequency histogram of the scores of all seventhgrade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS). The scores are grade-level equivalents. So a score of 6. 3 indicates that the student’s performance is typical for a student in the third month of grade 6. Describe the histogram: The histogram is roughly symmetric, and both tails fall off smoothly from a single center peak. There are no large gaps or obvious outliers. Suppose that we choose a Gary seventh-grader at random. Let X = the student’s ITBS vocabulary score. What’s P(X < 6. 0)? Statistics and Probability with Applications, 3 rd Edition 3

Continuous Random Variables It’s the probability that a randomly selected student, Gary, earned a vocabulary score less than the sixth-grade equivalent on the ITBS test. The area of the shaded bars in the relative frequency histogram in (a) represents the proportion of students with vocabulary scores less than 6. 0. How do we find this Area? ? Statistics and Probability with Applications, 3 rd Edition 4

Continuous Random Variables To find this area, we need to find an appropriate area formula for the shape of our histogram. To do this, we draw a smooth curve through the tops of the histogram bars as in figure (b). This is a good description of the overall pattern of the distribution. We call this curve a density curve. Statistics and Probability with Applications, 3 rd Edition 5

Continuous Random Variables Density Curve The probability distribution of a continuous random variable is described by a density curve, a curve that • is always on or above the horizontal axis, and • has an area of exactly 1 underneath it. Statistics and Probability with Applications, 3 rd Edition 6

Continuous random variables and density curves Suppose that a timer for a video game has a randomly determined length of time, after which an enemy appears. Let X = the length of time until an enemy appears. The probability distribution is shown as a density curve in the following figure. a) Is this graph a valid density curve? b) Find and interpret P(X < 1). Statistics and Probability with Applications, 3 rd Edition 7

TYPES OF DENSITY CURVES Since a Density Curve is an idealized description of the data, we classify the types of density curves by the shapes of the distributions they represent. Finding the area under the density curve is like finding the proportion of data values on a particular interval Statistics and Probability with Applications, 3 rd Edition 8

The time it takes for students to drive to school is evenly distributed with a minimum of 5 minutes and a range of 35 minutes. What is the height of the rectangle? a)Draw the distribution Where should the rectangle end? 1/35 5 Statistics and Probability with Applications, 3 rd Edition 40 9

b) What is the probability that it takes less than 20 minutes to drive to school? P(X < 20) = (15)(1/35) =. 4286 1/35 5 Statistics and Probability with Applications, 3 rd Edition 40 10

Uniform Distribution • Is a continuous distribution that is evenly (or uniformly) distributed • Has a density curve in the shape of a rectangle EX: The Citrus Sugar Company packs sugar in bags labeled 5 pounds. However, the packaging isn’t perfect and the actual weights are uniformly distributed with a mean of 4. 98 pounds and a range of. 12 pounds. Statistics and Probability with Applications, 3 rd Edition 11

a)Constructing the uniform distribution we draw a rectangle centered at the mean of 4. 98 extending. 06 in either direction. What is the height of this rectangle? What shape does a uniform distribution have? How long is this rectangle? 1/. 12 4. 92 Statistics and Probability with Applications, 3 rd Edition 4. 98 5. 04 12

• What is the probability that a randomly selected bag will weigh more than 4. 97 pounds? P(X > 4. 97) = . 07(1/. 12). 5833 What is =the length of the shaded region? 1/. 12 4. 92 Statistics and Probability with Applications, 3 rd Edition 4. 98 5. 04 13

• Find the probability that a randomly selected bag weighs between 4. 93 and 5. 03 pounds. P(4. 93<X<5. 03) = What is = the length of the shaded region? . 1(1/. 12). 8333 1/. 12 4. 92 Statistics and Probability with Applications, 3 rd Edition 4. 98 5. 04 14

Continuous Random Variables Recall that the mean is the “balance point” of a distribution. This figure illustrates this idea for the mean of a continuous random variable. Areas under a density curve represent probabilities. The median of a continuous random variable is the 50 th percentile of its probability distribution. Mean and Median of a Continuous Random Variable The mean of a continuous random variable is the point at which its probability distribution would balance if made of solid material. The median of a continuous random variable is the equal-areas point, the point that divides the area under the probability distribution in half. Statistics and Probability with Applications, 3 rd Edition 15

Continuous Random Variables The median of a symmetric probability distribution is at its midpoint. A symmetric density curve balances at its midpoint because the two sides are identical. So the mean and median of a continuous random variable with a symmetric probability distribution are equal. It isn’t so easy to spot the equal-areas point on a skewed probability distribution. In a right skewed distribution, the mean is greater than the median because the balance point of the distribution is pulled toward the long right tail. Statistics and Probability with Applications, 3 rd Edition 16

Mean versus median • The probability distribution of a continuous random variable is shown. Identify the location of the mean and median by letter. Justify your answers. Statistics and Probability with Applications, 3 rd Edition 17

LESSON APP 5. 5 Still waiting for the server? How does your Web browser get a file from the Internet? Your computer sends a request for the file to a Web server, and the Web server sends back a response. For one particular Web server, the time X (in seconds) after the start of an hour at which a randomly selected request is received has the uniform distribution shown in the figure. 1. What height must the density curve have to be valid? Justify your answer. 2. Find the probability that the request is received within the first 300 seconds (5 minutes) after the hour. 3. What is the mean of X ? Explain. 4. What is the median of X ? Explain. Statistics and Probability with Applications, 3 rd Edition 18

Continuous Random Variables Normal Distribution A normal distribution is described by a symmetric, single-peaked, bell-shaped density curve. Any normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. • All normal distributions have the same overall shape: symmetric, single-peaked, and bell-shaped. • The mean is located at the midpoint of the symmetric density curve and is the same as the median. • The standard deviation σ measures the variability (width) of a normal distribution. Statistics and Probability with Applications, 3 rd Edition 19

Graphing a normal distribution • Many studies on automobile safety suggest that when drivers of automobiles need to make emergency stops, the stopping distances follow an approximately normal distribution. Suppose that for one pickup truck traveling at 62 mph under typical conditions on dry pavement, the mean stopping distance is = 155 ft, with a standard deviation of = 3 ft. Sketch the probability distribution of X = the stopping distance for a randomly selected emergency stop. Label the mean and the points that are 1, 2, and 3 standard deviations from the mean. Statistics and Probability with Applications, 3 rd Edition 20

Continuous Random Variables Learning Targets After this lesson, you should be able to: ü Show that the probability distribution of a continuous random variable is valid and use the distribution to calculate probabilities. ü Determine the relative locations of the mean and median of a continuous random variable from the shape of its probability distribution. ü Draw a normal probability distribution with a given mean and standard deviation. Statistics and Probability with Applications, 3 rd Edition 21