1. Ver. 01082016 Chapter 5 Continuous Random Variables 1 Probability/Ch5

2. 2 W say that X is a continuous random variable if there Continuous random variable

3. 3 Probability/Ch5 As a result, Using the first fundamental theorem of calculus Example

4. 4 Probability/Ch5 Expectation & Variance

5. 5 Probability/Ch5

6. 6

7. 7 proof

8. 8 Probability/Ch5 Example Sol. Note that so, and A direct way is

9. 9 Probability/Ch5 A random variable is (standard) uniformly distributed if its density function f ( x ) follows A random variable X ~unif( α, β ) has the following p.d.f. and c.d.f.

10. 10 Probability/Ch5

11. 11 Probability/Ch5 Need to prove Normal (Gauss) distribution

12. 12 Probability/Ch5 Let We have It remains to show that

13. 13 Probability/Ch5

14. 14 Probability/Ch5 Taking differentiation,

15. 15 Probability/Ch5 Therefore,

16. 16 Probability/Ch5

17. 17 Probability/Ch5

18. 18 Probability/Ch5

19. 19 Probability/Ch5

20. 20 Probability/Ch5 Note that therefore,

21. 21 Probability/Ch5 That is or Recall Thus a exponentially distributed random variable is memoryless !!

22. 22 Probability/Ch5 So, and, So, A variation of exponential distribution is the Laplace distribution, which has the density function. So,

23. 23 Probability/Ch5 The meaning of the ‘hazard rate’ is suggested by the following: The hazard rate function uniquely determines the distribution function F. Why ?

24. 24 Probability/Ch5 Note that Thus and

25. 25 Probability/Ch5 However

26. 26 Probability/Ch5 where Note that

27. 27 Probability/Ch5 Suppose events are occurring randomly and in accordance with the three axioms for deriving Poisson distribution in sec.4.7. As a result, Taking differentiation, This shows that Here i.i.d stands for independent and identically distributed

28. 28 Probability/Ch5 Thus Since These can also be seen from the fact that

29. 29 Probability/Ch5 Example Sol.thus

30. 30 Probability/Ch5 The Cauchy distribution is an example of a distribution which has no mean, variance, or higher moments defined. However, Furthermore,

31. 31 Probability/Ch5 where

32. 32 Probability/Ch5

33. 33 Probability/Ch5 The left hand side

34. 34 Probability/Ch5

35. 35 Probability/Ch5 The distribution of a function of a random variable So,

36. 36 Probability/Ch5 So,Proof Example Sol.