1. Continuous Random Variable (1)

2. Discrete Random Variables Probability Mass Function (PMF)

3. Continuous Random Variable P[X=x]=0 Not possible to define a PMF for a continuous random variable

4. Discrete Random Variables Cumulative Distribution Function

5. PMF to CDF

6. Comparison Discrete RV: 1.Zero slope 2.Jumps in CDF Continuous RV: A continuous function

7. Slope of CDF function The slope at any point x indicates the probability that X is near x.

8. Probability Density Function (PDF) It is not possible to define a PMF function for a continuous variable because P[X=x]=0. We can, however, define a probability density function.

9. PDF of X

10. Example 3.3

11. Expected Value Discrete Random Variable

12. Example Find the expected stoppint point of the pointer

13. The Expected Value of a function Derived Discrete Random Variable Derived Continuous Random Variable Discrete Example

14. Variance and Standard Deviation

15. Key Points An average is a typical value of a random variable. The next question: – “What are the chances of observing an event far from the average?” The variance of a random variable X describes the difference between X and its expected value.

16. Definitions

17. Properties of Variance/Standard of Deviation

18. Discrete Example

19. Quiz 3.3