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

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Presentation transcript:

Ver Chapter 5 Continuous Random Variables 1 Probability/Ch5

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

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

4 Probability/Ch5 Expectation & Variance

5 Probability/Ch5

6

7 proof

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

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 Probability/Ch5

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

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

13 Probability/Ch5

14 Probability/Ch5 Taking differentiation,

15 Probability/Ch5 Therefore,

16 Probability/Ch5

17 Probability/Ch5

18 Probability/Ch5

19 Probability/Ch5

20 Probability/Ch5 Note that therefore,

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

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

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 Probability/Ch5 Note that Thus and

25 Probability/Ch5 However

26 Probability/Ch5 where Note that

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

29 Probability/Ch5 Example Sol.thus

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

31 Probability/Ch5 where

32 Probability/Ch5

33 Probability/Ch5 The left hand side

34 Probability/Ch5

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

36 Probability/Ch5 So,Proof Example Sol.