1. Continuous Random Variables Section 7.1.2

2. Starter 7.1.2 A 9-sided die has three faces that show 1, two faces that show 2, and one face each showing 3, 4, 5, 6. Let X be the number that shows face-up. Draw the PDF histogram of X.

3. Objectives Display the PDF of a continuous random variable as a density curve. Find the probability that a continuous random variable is within a specified interval by finding the area under a density curve. California Standard 4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

4. Definitions A random variable is a variable whose value is the numerical outcome of a random event. A continuous random variable can take on ANY value within a specified interval. The probability distribution function (PDF) of a continuous r.v. is represented by a density curve.

5. The Probability of a Continuous R.V. Consider the spinner on the overhead projector. When I spin it, what is the probability that it lands on a number between.3 and.5? What is the probability that it lands on a number between 0 and 1? Draw a density curve that shows the possible outcomes and probabilities. –You are drawing the PDF of X

6. The Density Curve of a Continuous R.V. For the spinner, you drew a rectangle whose X values go from 0 to 1 and whose Y values are always exactly 1. –Recall that this is called a uniform distribution The probability of an event is the area under the curve and above the range of X values that make up the event. –So P(.3  x .5) is 0.2 because that is the area under the curve above that range of x

7. The Probability of One Exact Outcome When I spin the spinner, what is the probability of hitting 0.3 exactly? –In notation: evaluate P(X = 3) No matter how close I come, a really precise measurement will show some error. Example: How tall are you EXACTLY? Example: Watch the TI try to match an exact number with its random number generator. –Run program CONTRV Conclusion: The probability of any exact outcome is always zero! –Note that the area above a point (like x =.3) is zero –So P(a  x  b) is the same as P(a < x < b)

8. A New Problem For a certain random variable the density curve that follows the line y = x –Draw the PDF of X (i.e. the density curve) You should have drawn a triangle that goes from the origin to (√2, √2) Find P(0 < x < 1) –.5(1)(1) =.5

9. Another Problem Suppose we know the heights of Northgate girls to be N(166, 7.3) cm –Remember what that means? Find P(height < 155) A normal distribution is a density curve, so draw a normal curve and find the area below h = 155 –Normalcdf(0, 155, 166, 7.3) =.0659 –So P(h < 155) = 6.6% –We could also find z = (155 – 166) / 7.3 = -1.51 and use Table A

10. Objectives Display the PDF of a continuous random variable as a density curve. Find the probability that a continuous random variable is within a specified interval by finding the area under a density curve. California Standard 4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

11. Homework Read pages 375 – 378 Do problems 4 and 5