Functions of Random Variables & Expectation, Mean and Variance

Transcription

1 Functions of Random Variables & Expectation, Mean and Variance Kuan-Yu Chen ( 陳冠宇 TR-409, NTUST

2 Functions of Random Variables 1 Given a random variables XX, one may generate other random variables by applying various transformations on XX YY = gg(xx) The transformation can be either linear or non-linear Let the random variable XX be the temperature in degrees Celsius, and consider the transformation YY = 1.8XX + 32, which gives the temperature in degrees Fahrenheit YY = aa XX + bb If we wish to display temperatures on a logarithmic scale YY = log(xx) 2

3 Functions of Random Variables 2 If YY = gg(xx) is a function of a random variable XX, then YY is also a random variable, since it provides a numerical value for each possible outcome Every outcome in the sample space defines a numerical value for XX and hence also the numerical value yy = gg() for YY To obtain pp YY (yy) for any yy, we add the probabilities of all values of such that gg = yy pp YY yy = { gg =yy} pp XX pp XX () pp YY (yy) pp YY yy 1 = pp XX 1 + pp XX 3 pp YY yy 2 = pp XX 2 Sample Space yy 1 yy 2 yy 3

4 Examples 1 pp YY 0 = pp XX 0 = 1 9 pp YY 1 = pp XX 1 + pp XX 1 = 2 9 pp YY 2 = pp XX 2 + pp XX 2 = 2 9 pp YY yy = 2, iiii yy = 1, 2, 3, 4 9 1, iiii yy = 0 9 0, ooooooooooooooooo pp YY yy = { gg =yy} pp XX 4

5 Examples 2 pp ZZ zz = { zz= 2 } pp XX pp ZZ zz = 2, iiii zz = 1, 4, 9, , iiii yy = 0 9 0, ooooooooooooooooo 5

6 Expectation 1 An Illustrative Example: Suppose that you spin the wheel kk times, and that kk ii is the number of times that the outcome (money) is mm ii (there are nn distinct outcomes, mm 1, mm 2,, mm nn ) kk 1 + kk kk nn = kk The total amount received is mm 1 kk 1 + mm 2 kk mm nn kk nn The amount received per spin is mm 1 kk 1 + mm 2 kk mm nn kk nn kk 6

7 Expectation 2 If the number of spins kk is very large, and if we are willing to interpret probabilities as relative frequencies, it is reasonable to anticipate that mm ii comes up a fraction of times that is roughly equal to pp ii pp ii kk ii kk By doing so, the amount received per spin can be also represented as The money you expect to get per spin mm 1 kk 1 + mm 2 kk mm nn kk nn kk mm 1 pp 1 + mm 2 pp mm nn pp nn 7

8 Expectation 3 It can be viewed as the center of gravity 8

9 Example pp XX = 1 4 1, iiii = , iiii = , iiii = 2 4 EE XX = = = 3 2 9

10 Moments The nn th moment of a random variable XX is the expected value of a random variable XX nn EE XX nn = nn pp XX () The first moment By the definition, the first moment of XX is just the mean EE XX = pp XX () The second moment EE XX 2 = 2 pp XX () 10

11 Variance & Standard Deviation The most important quantity associated with a random variable XX, other than the mean, is its variance var XX = EE XX EE[XX] 2 Since XX EE[XX] 2 can only take nonnegative values, the variance is always nonnegative The variance provides a measure of dispersion of XX around its mean Another measure of dispersion is the standard deviation It is defined as the square root of the variance σσ XX = var XX = EE XX EE[XX] 2 11

12 Example Consider a random variable XX,which has the PMF 1 pp XX =, 9 iiii iiii aaaa iiiiiiiiiiiiiiii iiii ttttt rrrrrrrrrr [ 4,4] 0, ooooooooooooooooo The mean: EE XX = The variance: = 0 llllll ZZ = XX EE(XX) 2 = XX 2 pp ZZ zz = EE XX EE(XX) 2 = EE ZZ = = , iiii zz = 1, 4, 9, , iiii zz = 0 9 0, ooooooooooooooooo 12

13 Functions of Random Variables 4 EE gg(xx) = 1 pp XX =, 9 iiii iiii aaaa iiiiiiiiiiiiiiii iiii ttttt rrrrrrrrrr [ 4,4] 0, ooooooooooooooooo = 4 gg(xx) = XX EE(XX) 2 gg()pp XX () = ( 4) ( 3) ( 2) ( 1) = 60 9 = EE XX EE(XX) 2 13

14 Summary 14

15 Properties of Mean and Variance 1 EE YY = EE aaaa + bb = (aaaa + bb) pp XX () = aaaa pp XX () + bb pp XX () = aa pp XX () + bb pp XX () = aaee XX + bb 15

16 Properties of Mean and Variance 2 var YY = EE YY EE[YY] 2 = EE aaxx + bb EE[aaXX + bb] 2 = aa + bb EE[aaXX + bb] 2 pp XX () = aa + bb aaee XX bb 2 pp XX () = = aa 2 EE XX 2 pp XX () = aa 2 var(xx) aa EE XX 2 pp XX () 16

17 Properties of Mean and Variance 3 var XX = EE[XX] 2 pp XX () = ( 2 2EE XX + EE[XX] 2 ) pp XX () = 2 pp XX () 2EE XX pp XX + = EE XX 2 2EE XX pp XX + EE XX = EE XX 2 2 EE XX 2 + EE XX 2 = EE XX 2 EE XX 2 2 EE XX pp XX 2 pp XX 17

18 Examples 1 18

19 Examples 2 19

20 Examples 3. 20

21 Examples

22 Examples 4 pp TT tt = 0.6, iiii tt = , iiii tt = 2 30 EE TT = =

23 Questions? 23