University:
University of Houston
Course:
MATH 2312 | Precalculus
Academic year:

2022
Views:

203

Pages:

14
Author:

Charlize Mccoy

Density Curves Section 4.1 Outline 1 Beginning Questions 2 Continuous Probability Distribution 3 Examples of Density Curves 4 Uniform Distribution 5 Shapes of Density Curves Popper Set Up Fill in all of the proper bubbles. Use a #2 pencil. This is popper number 05. Popper Questions 1. A sample is the set of all possible data values for a given subject under consideration. a. True b. False 2. X is the number of days it rained last month where you lived. X is an example of a discrete random variable. a. True b. False 3. X is the amount of rainfall in your state last month. X is an example of a continuous random variable. a. True b. False Probability Distribution The probability distribution of a random variable X tells us what values X can take and how to assign probabilities to those values. Requirements for a probability distribution: 1. The sum of all the probabilities equal 1. 2. The probabilities are between 0 and 1, including 0 and 1. Continuous Probability Distribution The probability distribution of a continuous random variable X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that makes up the event. The mean is the center or expected value of that distribution. The standard deviation is the spread of that distribution. Density Curves A mathematical model for a probability distribution of a continuous random variable. This curve is always on or above the horizontal axis. The area under a density curve is exactly 1. The area under the curve and between any range of values is the probability that an observation falls in that range. Example of Density Curve The following graph is an example of a density curve that consists of two line segments. The first goes from the point (0,1) to the point (0.4,1). The second goes from (0.4,1) to (0.8,2) in the xy-plane. Does this meet the requirements of a probability distribution? What percent of the observations fall below 0.4? What percent of the observations lie between 0.4 and 0.8? What percent of observations are equal to 0.4? Example of a density curve: Uniform distribution The Uniform Distribution describes a variable that takes values that are uniformly spread between a range of values. Thus it takes on a rectangular shape. The proportion (percent) of observations that lie within a range of values is equivalent to the area of the rectangle between the desired range of values. Area = Height × Width The height of the rectangle is 1 highest value−lowest value . Uniform distribution The following histogram is of waiting times for an elevator where the longest waiting time is 5 minutes. Histogram of Waiting Times 0.25 Density 0.20 0.15 0.10 0.05 0.00 0 1 2 3 Waiting Times 4 5 Uniform distribution example The random variable X is the waiting time for the elevator. The possible values are 0 ≤ X ≤ 5. Since we are looking at a random variable that assumes values corresponding to an interval this is a continuous random variable. The probabilities for this random variable is the same as the area under a density curve. In this example any one of the times has an equally likely chance of assuming a value between 0 and 5. Thus this curve is rectangular.