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# Untitled

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1.
2. ## Calculation labs
3.
4. choose(4,2)
5. factorial(5)
6. #EX 1
7. choose(8,3)*factorial(3)
8. #EX 2
9. # a)
10. #permutation because order matters
11. choose(36,8)*factorial(8)
12. factorial(36)/factorial(36-8)
13. # b)
14. #if letters can repeat
15. 36^8
16. #EX 3
17. #combination because order doesnt matter (n=10, r =4)
18. choose(10,4)
19. #EX 4
20. # P(all def.) = 1 (in mean choose(3,3)) / choose(14,3)
21. 1/choose(14,3)
22. # also
23. 3/14*2/13*1/12
24. #EX 5
25. choose(6,6)/choose(49,6)
26. choose(6,5)*choose(43,1)/choose(49,6)
27. choose(6,4)*choose(43,2)/choose(49,6)
28. choose(6,3)*choose(43,3)/choose(49,6)
29. choose(6,2)*choose(43,4)/choose(49,6)
30. choose(6,1)*choose(43,5)/choose(49,6)
31. choose(6,0)*choose(43,6)/choose(49,6)
32.
33. probs <- choose(6,0:6)*choose(43,6-0:6)/choose(49,6)
34. plot(probs)
35. format(probs, digits = 6, format="f")
36.
37. ## Bayesian revision labs
38.
39. # EX 1
40. prior<-c(.3,.5,.2)
41. likelyhood<-c(.7,.4,.2)
42. posterior_a<-prior*likelyhood
43. posterior<-posterior_a/sum(posterior_a)
44.
45. #EX 2
46. prior<-c(.9,.1)
47. likelyhood<-c(.98,.05)
48. posterior_a<-prior*likelyhood
49. posterior<-posterior_a/sum(posterior_a)
50.
51. #EX 3
52. prior<-c(1/6,1/6,1/6,1/6,1/6,1/6)
53. likelyhood<-c(0,.5,.5,1,1,1)
54. posterior_a<-prior*likelyhood
55. posterior<-posterior_a/sum(posterior_a)
56.
57. #EX 4
58. prior<-c(1/3,1/3,1/3)
59. likelyhood<-c(0,0.5,1)
60. posterior_a<-prior*likelyhood
61. posterior<-posterior_a/sum(posterior_a)
62.
63. # Discrete random variables
64.
65. #EX 1
66. x<-1:6
67. prob<-rep(1/6,6)
68. #expected value
69. expected_value<-sum(x*prob)
70. #variance
71. variance<-sum((x-expected_value)^2*prob)
72.
73. #EX 2
74. n<-3
75. x<-0:n
76. #prob<-c(.81,.18,.01)
77. binomial_distribution<-dbinom(x,n,.1)
78. #expected value
79. expected_value<-sum(x*binomial_distribution)
80. #variance
81. variance<-sum((x-expected_value)^2*binomial_distribution)
82.
83. #expected value = n * p
84. #variance = n * p * q
85.
86. #EX 5
87. x<-c(0,0,0,24,200,5000,9000000)-3
88. prob<-(dhyper(0:6,6,43,6))
89. #expected value
90. expected_value<-sum(x*prob)
91. #variance
92. variance<-sum((x-expected_value)^2*prob)
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