1. Continuous Random Variables

3. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of times that category occurred.

4. Probability Density Function As more and more different categories occur the “bars” get thinner and thinner If there are an infinite number of categories, the bars are infinitesimally wide

5. Probability density function

7. Uniform distribution

8. Numerical Integration in R The integrate() function is used to numerically integrate functions in R.

9. Example

11. Numerical Integration in R The integrate() function is used to numerically integrate functions in R.

12. The cumulative distribution function

13. Computing probabilities using the cdf

14. Fig. 4-8, p. 138 F(b) F(a) F(b) - F(a)

15. Example

16. Percentiles of a continuous distribution

17. Fig. 4-10, p. 139

18. Quantiles in R In R, all of the built in distributions have a built in function called the quantile function which calculates percentiles. The quantile function always begins with the letter q. So for instance: Suppose that Z has a standard normal distribution(to be introduced soon) and we wish to determine the 74 th percentile of Z, i.e. the value pp such that P(Z < pp) =.74. In R we just use the qnorm() function as follows: So P(Z<.6433454) ~.74 To verify in R:

19. Mean of a continuous random variable

20. Expected value of a function of a rv

21. Variance of a continuous rv

22. Example Compute the mean of this rv Compute the standard deviation of this rv

23. Example Compute the mean of this rv

24. Example Compute the standard deviation of this rv