概率論簡介PPT課件

1、.,1,1,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456,This sequence provides an example of a discrete random variable. Suppose that you have a red die which, when thrown, takes the numbers from 1 to 6 with equal probability.,.,2,2,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM O

2、F TWO DICE,red123456 green 1 2 3 4 5 6,Suppose that you also have a green die that can take the numbers from 1 to 6 with equal probability.,.,3,3,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1 2 3 4 5 6,We will define a random variable X as the sum of the numbers when t

3、he dice are thrown.,.,4,4,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1 2 3 4 5 610,For example, if the red die is 4 and the green one is 6, X is equal to 10.,.,5,5,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,Similarly, if the red die is 2 and the green

4、one is 5, X is equal to 7.,red123456 green 1 2 3 4 57 6,.,6,6,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112,The table shows all the possible outcomes.,.,7,7,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,r

5、ed123456 green 1234567 2345678 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,If you look at the table, you can see that X can be any of the numbers from 2 to 12.,.,8,8,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DI

6、CE,red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,We will now define f, the frequencies associated with the possible values of X.,.,9,9,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DIC

7、E,red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,For example, there are four outcomes which make X equal to 5.,.,10,10,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green

8、 1234567 2345678 3456789 45678910 567891011 6789101112,Similarly you can work out the frequencies for all the other values of X.,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,.,11,11,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1234

9、567 2345678 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,Finally we will derive the probability of obtaining each value of X.,.,12,12,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1234567 234567

10、8 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36,If there is 1/6 probability of obtaining each number on the red die, and the same on the green die, each outcome in the table will occur with 1/36 probability.,.,13,13,Hence to

11、 obtain the probabilities associated with the different values of X, we divide the frequencies by 36.,PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE,red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112,Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1

12、122/36 1211/36,.,14,14,The distribution is shown graphically. in this example it is symmetrical, highest for X equal to 7 and declining on either side.,6 _ 36,5 _ 36,4 _ 36,3 _ 36,2 _ 36,2 _ 36,3 _ 36,5 _ 36,4 _ 36,probability,2,3,4,5,6,7,8,9,10,11,12,X,1 36,1 36,PROBABILITY DISTRIBUTION EXAMPLE: X

13、IS THE SUM OF TWO DICE,.,15,EXPECTED VALUE OF A RANDOM VARIABLE,1,The expected value of a random variable, also known as its population mean, is the weighted average of its possible values, the weights being the probabilities attached to the values.,Definition of E(X), the expected value of X:,.,16,

14、2,Note that the sum of the probabilities must be unity, so there is no need to divide by the sum of the weights.,EXPECTED VALUE OF A RANDOM VARIABLE,Definition of E(X), the expected value of X:,.,17,3,EXPECTED VALUE OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3

15、x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),This sequence shows how the expected value is calculated, first in abstract and then with the ran

16、dom variable defined in the first sequence. We begin by listing the possible values of X.,.,18,4,EXPECTED VALUE OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p

17、894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),Next we list the probabilities attached to the different possible values of X.,.,19,5,EXPECTED VALUE OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36

18、x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),Then we define a column in which the values are weighted by the corresponding probabilities.,.,20,6,EXPECTED VALU

19、E OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),We do this

20、for each value separately.,.,21,7,EXPECTED VALUE OF A RANDOM VARIABLE,Here we are assuming that n, the number of possible values, is equal to 11, but it could be any number.,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6

21、p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),.,22,8,EXPECTED VALUE OF A RANDOM VARIABLE,The expected value is the sum of the entries in the third column.,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232

22、/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),.,23,9,EXPECTED VALUE OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2

23、p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),The random variable X defined in the previous sequence could be any of the i

24、ntegers from 2 to 12 with probabilities as shown.,.,24,10,EXPECTED VALUE OF A RANDOM VARIABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10

25、 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),X could be equal to 2 with probability 1/36, so the first entry in the calculation of the expected value is 2/36.,.,25,11,EXPECTED VALUE OF A RANDOM VARIABLE,The probability of x being equal to 3 was 2/36, so the second entry is 6/36.,xipix

26、i pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),.,26,12,EXPECTED VALUE OF A RANDOM VARI

27、ABLE,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X),Similarly for the other 9 pos

28、sible values.,.,27,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X) 252/36,13,To ob

29、tain the expected value, we sum the entries in this column.,EXPECTED VALUE OF A RANDOM VARIABLE,.,28,xipixi pixi pi xi pi x1p1x1 p121/362/36 x2p2x2 p232/366/36 x3p3x3 p343/3612/36 x4p4x4 p454/3620/36 x5p5x5 p565/3630/36 x6p6x6 p676/3642/36 x7p7x7 p785/3640/36 x8p8x8 p894/3636/36 x9p9x9 p9103/3630/36

30、 x10p10 x10 p10112/3622/36 x11p11x11 p11121/3612/36 S xi pi = E(X) 252/36 = 7,14,EXPECTED VALUE OF A RANDOM VARIABLE,The expected value turns out to be 7. Actually, this was obvious anyway. We saw in the previous sequence that the distribution is symmetrical about 7.,.,29,Very often the expected val

31、ue of a random variable is represented by m, the Greek m. If there is more than one random variable, their expected values are differentiated by adding subscripts to m.,15,EXPECTED VALUE OF A RANDOM VARIABLE,Alternative notation for E(X):,.,30,Definition of , the expected value of a function of X:,1

32、,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,To find the expected value of a function of a random variable, one calculates all the possible values of the function, weights them by the corresponding probabilities, and sums the results.,.,31,Definition of , the expected value of a function of X:

33、 Example:,For example, the expected value of X2 is found by calculating all its possible values, multiplying them by the corresponding probabilities, and summing.,2,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,.,32,3,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2

34、xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,The calculation of the expected value of a function of

35、a random variable will be outlined in general and then illustrated with an example.,.,33,4,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/3649

36、8.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,First one makes a list of the possible values of X and the corresponding probabilities.,.,34,5,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(

37、x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,Next the function of X is calculated for each possible value of X.,.

38、,35,6,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn12

39、1/361444.00 S g(xi) pi 54.83,Then, one at a time, the value of the function is weighted by its corresponding probability.,.,36,7,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33

40、 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,This is done individually for each possible value of X.,.,37,8,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi

41、x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,The sum of the weighted values is the expected value of the fu

42、nction of X.,.,38,9,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(

43、xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,The process will be illustrated for X2, where X is the random variable defined in the first sequence. The 11 possible values of X and the corresponding probabilities are listed.,.,39,10,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi

44、 pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,First one calculates the possible values of X2.,

45、.,40,11,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn

46、121/361444.00 S g(xi) pi 54.83,The first value is 4, which arises when X is equal to 2. The probability of X being equal to 2 is 1/36, so the weighted function is 4/36, which we shall write in decimal form as 0.11.,.,41,12,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,xipig(xi) g(xi ) pixi pi xi

47、2xi2 pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,Similarly for all the other possible values of X.,.,42

48、,13,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,The expected value of X2 is the sum of its weighted values in the final column. It is equal to 54.83. It is the average value of the figures in the previous column, taking the differing probabilities into account.,xipig(xi) g(xi ) pixi pi xi2xi2

49、pi x1p1g(x1)g(x1) p121/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,.,43,xipig(xi) g(xi ) pixi pi xi2xi2 pi x1p1g(x1)g(x1) p121

50、/3640.11 x2p2g(x2) g(x2) p232/3690.50 x3p3g(x3) g(x3) p343/36161.33 . 54/36252.78 . 65/36365.00 . 76/36498.17 . 85/36648.89 . 94/36819.00 . 103/361008.83 . 112/361216.72 xnpng(xn) g(xn) pn121/361444.00 S g(xi) pi 54.83,Note that E(X2) is not the same thing as E(X), squared. In the previous sequence

51、we saw that E(X) for this example was 7. Its square is 49.,14,EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE,.,44,1.,1,EXPECTED VALUE RULES,This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two random variables is the sum o

52、f their expected values.,.,45,1.,2,EXPECTED VALUE RULES,Here the sum consists of three variables. But the rule generalizes to any number.,.,46,1.,3,EXPECTED VALUE RULES,2.,The second rule is the multiplicative rule. The expected value of a variable that has been multiplied by a constant) is equal to

53、 the constant multiplied by the expected value of the variable.,.,47,1.,4,EXPECTED VALUE RULES,2.,For example, the expected value of 3X is three times the expected value of X.,Example:,.,48,1.,5,EXPECTED VALUE RULES,2.,3.,Finally, the expected value of a constant is just the constant. Of course this

54、 is obvious.,.,49,1.,6,EXPECTED VALUE RULES,2.,3.,As an exercise, we will use the rules to simplify the expected value of an expression. Suppose that we are interested in the expected value of a variable Y, where Y = b1 + b2X.,.,50,1.,7,EXPECTED VALUE RULES,2.,3.,We use the first rule to break up th

55、e expected value into its two components.,.,51,Then we use the second rule to replace E(b2X) by b2E(X) and the third rule to simplify E(b1) to just b1. This is as far as we can go in this example.,1.,8,EXPECTED VALUE RULES,2.,3.,.,52,The third sequence defined the expected value of a function of a r

56、andom variable X. There is only one function that is of much interest to us, at least initially: the squared deviation from the population mean.,1,POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE,Population variance of X:,.,53,The expected value of the squared deviation is known as the population variance of X. It is a measure of the dispers