1. PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Two Random Variables

2. Introduction  Expressing observations using more than one quantity -height and weight of each person -number of people and the total income in a family

3. Joint Probability Distribution Function  Let X and Y denote two RVs based on a probability model ( , F, P). Then:  What about  So, Joint Probability Distribution Function of X and Y is defined to be

4. Properties(1) Proof   Since

5. Properties(2) Proof   for  Since this is union of ME events,  Proof of second part is similar.

6. Properties(3) Proof   So, using property 2, we come to the desired result.

7. Joint Probability Density Function (Joint p.d.f)  By definition, the joint p.d.f of X and Y is given by  Hence,  So, using the first property,

8. Calculating Probability  How to find the probability that ( X,Y ) belongs to an arbitrary region D ?  Using the third property,

9. Marginal Statistics  Statistics of each individual ones are called Marginal Statistics.  marginal PDF of X,  the marginal p.d.f of X.  Can be obtained from the joint p.d.f. Proof

10. Marginal Statistics Using the formula for differentiation under integrals, t aking derivative with respect to x we get: Proof

11.  Let  Then: Differentiation Under Integrals

12. Marginal p.d.fs for Discrete r.vs  X and Y are discrete r.vs  Joint p.d.f:  Marginal p.d.fs:  When written in a tabular fashion, to obtain one needs to add up all entries in the i-th row.  This suggests the name marginal densities.

13. Example  Given marginals, it may not be possible to compute the joint p.d.f.  Obtain the marginal p.d.fs and for: Example

14.  is constant in the shaded region  We have: So:  Thus c = 2. Moreover,  Similarly,  Clearly, in this case given and, it will not be possible to obtain the original joint p.d.f in Solution

15. Example  X and Y are said to be jointly normal (Gaussian) distributed, if their joint p.d.f has the following form:  By direct integration,  So the above distribution is denoted by Example

16. Example - continued  So, once again, knowing the marginals alone doesn’t tell us everything about the joint p.d.f  We will show that the only situation where the marginal p.d.fs can be used to recover the joint p.d.f is when the random variables are statistically independent.

17. Independence of r.vs The random variables X and Y are said to be statistically independent if the events and are independent events for any two Borel sets A and B in x and y axes respectively.  For the events and if the r.vs X and Y are independent, then  i.e.,  or equivalently:  If X and Y are discrete-type r.vs then their independence implies Definition

18. Independence of r.vs  Given  obtain the marginal p.d.fs and  examine whether independence condition for discrete/continuous type r.vs are satisfied.  Example: Two jointly Gaussian r.vs as in (7-23) are independent if and only if the fifth parameter Procedure to test for independence

19. Example Given Determine whether X and Y are independent. Similarly In this case and hence X and Y are independent random variables. Solution