1. , t, F distributions

2. distribution
t distribution
F distribution 2

3. You don’t need to care about complicated math formula.
Just see, and remember only what I marked with

4. iid = independent and identically distributed★
Random sample (r.s.) :
A sample which are collected by independent sampling procedure from an identically distributed population.
iid = independent and identically distributed

5. X\Y 1 2 T
1/16
1/8
1/4
1/2
1.0
X\Y 1 2 T
1/12
1/4
1/6
1/2
1/3
1.0
X\Y 1 2 3 T
1/16
1/8
1/4
1/2
1.0
X\Y 1 2 T
1/4
1/2
1.0

6. are (mutually) independent (a)
are iid
are (mutually) independent
(a)
(b) 6

7. 7

8. 8

9. 9

10. : degree of freedom ( a parameter, related to number of var’s)10

11. 11

13. 13

14. ★ 14

15. ★ 15

16.  2 ; n  2 ; n 1 3 5 8 15 24 ∞
0.975
0.001
0.216
0.831
2.180
6.262
12.401
0.95
0.004
0.352
1.145
2.733
7.261
13.848
0.05
3.841
7.815
11.071
15.507
24.996
36.415
0.025
5.024
9.348
12.833
17.535
27.488
39.364 ∞ ∞ ∞

17. For large 17

18. Lorentzian
Laplacian
Normal
Cauchy
Double
Expon.
Gaussian

19. -4 -2 2 4
0.0
0.1
0.2
0.3
0.4
0.5

20. Louis F. Cauchy
(1760–1848)
Hendrik A. Lorentz
(1853–1928)

21. 21

22. : degree of freedom ( a parameter, related to number of var’s)22

23. ★ 23

24. ★ Cauchy N(0,1) N(0,1) t(30) t(1) t(1), t(2), t(4), t(30) t(1) t(1)
t(n
t(n ) ) t( t( ∞ ∞ ) )
Cauchy
N(0,1)
0.4
N(0,1)
t(30)
0.3
0.2
t(1)
0.1
0.0 -4 -2 2 4
t(1), t(2), t(4), t(30)

25. 0.3
1.0
0.8
0.2
0.6
0.4
0.1
0.2
0.0
0.0
-15
-10 -5 5 10 15
-15
-10 -5 5 10 15 1 3 8 15 21 23 ∞
0.10
3.078
1.638
1.397
1.341
1.323
1.319
1.282
0.05
6.314
2.353
1.860
1.753
1.721
1.714
1.645
0.025
12.706
3.182
2.306
2.131
2.080
2.069
1.96

26. Student (1908), The probable error of a mean. Biometrika, 6-1, 1-25.
William Sealy Gosset ( )
Student (1908), The probable error of a mean. Biometrika, 6-1, 1-25.

28. : degree of freedom 28

29. Variance of F distribution is complicated.
for large
Variance of F distribution is complicated.
You don’t need to care about these. 29

30. Most of cases the center of dist’n is near 1.1 2 3 4 5 6
0.0
0.5
1.0
1.5
Most of cases the center of dist’n is near 1.

31. George W. Snedecor ( )
R. A. Fisher (1890 –1962)

32. 32

34. 1 2 3 4 5 6 7
0.1
0.3
0.5
0.7

35. 1 2 6 8 9 15
161.45
18.513
5.987
5.318
5.117
4.543
199.50
19.000
5.143
4.459
4.256
3.682
233.99
19.330
4.284
3.581
3.374
2.790
238.88
19.371
4.147
3.438
3.230
2.641
240.54
19.385
4.099
3.388
3.179
2.588
245.95
19.429
3.938
3.218
3.006
2.403 1 2 6 8 9 15
647.79
38.506
8.813
7.571
7.209
6.200
799.50
39.000
7.260
6.059
5.715
4.765
937.11
39.331
5.820
4.652
4.320
3.415
956.66
39.373
5.600
4.433
4.102
3.199
963.29
39.387
5.523
4.357
4.026
3.123
984.87
39.431
5.269
4.101
3.769
2.862

36. ★ 36

37. 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7
0.358
3.94
0.254
2.79

38. 38

39. -4 -2 2 4 2 4 6 8
-2.447
2.447

40. Thank you !!