Given P(x)
     x           P(x)       x*P(x)      (x-µ)       (x-µ)2    p(x)*(x-µ)2     Mean
             1     0.2700   ...
Find P(x)       Sum
                      1000
     x          Frequency    P(x)          x*P(x)      (x-µ)       (x-µ)2  ...
At most (i.e.
                                       less than 3
                                        means at
Binomial...
55   0.0000    1.0000    0.0000
 56   0.0000    1.0000    0.0000
 57   0.0000    1.0000    0.0000
 58   0.0000    1.0000  ...
Geometric                                         µ = 1/p        σ^2 = q/p^2
P(x) = pq^(x-1)          x        P(x)       ...
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Chapter 4

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Chapter 4

  1. 1. Given P(x) x P(x) x*P(x) (x-µ) (x-µ)2 p(x)*(x-µ)2 Mean 1 0.2700 0.27 -1.54 2.37 0.64 2.54 2 0.3100 0.62 -0.54 0.29 0.09 Variance 3 0.1800 0.54 0.46 0.21 0.04 1.87 4 0.0900 0.36 1.46 2.13 0.19 Standard Dev 5 0.1500 0.75 2.46 6.05 0.91 1.37 Expected Value 2.54
  2. 2. Find P(x) Sum 1000 x Frequency P(x) x*P(x) (x-µ) (x-µ)2 p(x)*(x-µ)2 Mean 0 316 0.32 0 -1.09 1.19 0.38 1.09 1 425 0.43 0.43 -0.09 0.01 0 Variance 2 168 0.17 0.34 0.91 0.83 0.14 1.15 3 48 0.05 0.14 1.91 3.64 0.17 Standard Dev 4 29 0.03 0.12 2.91 8.46 0.25 1.07 5 14 0.01 0.07 3.91 15.28 0.21 Expected Value 1.09
  3. 3. At most (i.e. less than 3 means at Binomial Dist x Exact most 2) At least 0 0.60022 n 1 0.30767 0.9079 0.39978252 61 2 0.07757 0.9855 0.0921 p 3 0.01282 0.9983 0.0145430 0.01 4 0.001561963 0.9998 0.0017 q 5 0.00015 1.0000 0.000162190 0.99 6 0.00001 1.0000 0.0000 7 0.00000 1.0000 0.0000 8 0.00000 1.0000 0.0000 Mean 9 0.00000 1.0000 0.0000 0.50833 10 0.00000 1.0000 0.0000 Variance 11 0.00000 1.0000 0.0000 0.5 12 0.00000 1.0000 0.0000 Standard Deviation 13 0.00000 1.0000 0.0000 0.71 14 0.00000 1.0000 0.0000 15 0.00000 1.0000 0.0000 16 0.00000 1.0000 0.0000 17 0.00000 1.0000 0.0000 18 0.00000 1.0000 0.0000 19 0.00000 1.0000 0.0000 20 0.00000 1.0000 0.0000 21 0.000000000 1.0000 0.0000 22 0.000000000 1.0000 0.0000 23 0.0000 1.0000 0.0000 24 0.0000 1.0000 0.0000 25 0.0000 1.0000 0.0000 26 0.0000 1.0000 0.0000 27 0.0000 1.0000 0.0000 28 0.0000 1.0000 0.0000 29 0.0000 1.0000 0.0000 30 0.0000 1.0000 0.0000 31 0.0000 1.0000 0.0000 32 0.0000 1.0000 0.0000 33 0.0000 1.0000 0.0000 34 0.0000 1.0000 0.0000 35 0.0000 1.0000 0.0000 36 0.0000 1.0000 0.0000 37 0.0000 1.0000 0.0000 38 0.0000 1.0000 0.0000 39 0.0000 1.0000 0.0000 40 0.0000 1.0000 0.0000 41 0.0000 1.0000 0.0000 42 0.0000 1.0000 0.0000 43 0.0000 1.0000 0.0000 44 0.0000 1.0000 0.0000 45 0.0000 1.0000 0.0000 46 0.0000 1.0000 0.0000 47 0.0000 1.0000 0.0000 48 0.0000 1.0000 0.0000 49 0.0000 1.0000 0.0000 50 0.0000 1.0000 0.0000 51 0.0000 1.0000 0.0000 52 0.0000 1.0000 0.0000 53 0.0000 1.0000 0.0000 54 0.0000 1.0000 0.0000
  4. 4. 55 0.0000 1.0000 0.0000 56 0.0000 1.0000 0.0000 57 0.0000 1.0000 0.0000 58 0.0000 1.0000 0.0000 59 0.0000 1.0000 0.0000 60 0.0000 1.0000 0.0000 61 0.0000 1.0000 0.0000 62 Err:502 Err:502 0.0000 63 Err:502 Err:502 Err:502 64 Err:502 Err:502 Err:502 65 Err:502 Err:502 Err:502 66 Err:502 Err:502 Err:502 67 Err:502 Err:502 Err:502 68 Err:502 Err:502 Err:502 69 Err:502 Err:502 Err:502 70 Err:502 Err:502 Err:502 71 Err:502 Err:502 Err:502 72 Err:502 Err:502 Err:502 73 Err:502 Err:502 Err:502 74 Err:502 Err:502 Err:502 75 Err:502 Err:502 Err:502 76 Err:502 Err:502 Err:502 77 Err:502 Err:502 Err:502 78 Err:502 Err:502 Err:502 79 Err:502 Err:502 Err:502 80 Err:502 Err:502 Err:502 81 Err:502 Err:502 Err:502 82 Err:502 Err:502 Err:502 83 Err:502 Err:502 Err:502 84 Err:502 Err:502 Err:502 85 Err:502 Err:502 Err:502 86 Err:502 Err:502 Err:502 87 Err:502 Err:502 Err:502 88 Err:502 Err:502 Err:502 89 Err:502 Err:502 Err:502 90 Err:502 Err:502 Err:502 91 Err:502 Err:502 Err:502 92 Err:502 Err:502 Err:502 93 Err:502 Err:502 Err:502 94 Err:502 Err:502 Err:502 95 Err:502 Err:502 Err:502 96 Err:502 Err:502 Err:502 97 Err:502 Err:502 Err:502 98 Err:502 Err:502 Err:502 99 Err:502 Err:502 Err:502 100 Err:502 Err:502 Err:502 101 Err:502 Err:502 Err:502 102 Err:502 Err:502 Err:502 103 Err:502 Err:502 Err:502 104 Err:502 Err:502 Err:502 105 Err:502 Err:502 Err:502 106 Err:502 Err:502 Err:502 107 Err:502 Err:502 Err:502 108 Err:502 Err:502 Err:502 109 Err:502 Err:502 Err:502 110 Err:502 Err:502 Err:502 111 Err:502 Err:502 Err:502 112 Err:502 Err:502 Err:502 113 Err:502 Err:502 Err:502
  5. 5. Geometric µ = 1/p σ^2 = q/p^2 P(x) = pq^(x-1) x P(x) µ σ^2 S.d p 1 0.0800 12.5 143.75 11.99 0.08 2 0.07 q 3 0.07 0.14 0.92 4 0.06 5 0.06 6 0.05 7 0.05 8 0.04 0.09 9 0.04 16 0.02 Poisson x µ P(x) var=µ P(x) = (µ^xe^-µ)/x! 0 12 0 12 1 12 0 s.d 2 12 0 3.46 3 12 0 4 12 0.01 5 12 0.01 6 12 0.03 0.99 7 12 0.04 8 12 0.07 9 12 0.09 10 12 0.1 11 12 0.11 12 12 0.11 13 12 0.11 14 12 0.09048890 15 12 0.07239112 16 12 0.05429334 17 12 0.03832471 18 12 0.02554981 19 12 0.01613672 20 12 0.00968203 21 12 0.00553259 22 12 0.00301778 23 12 0.00157449 24 12 0.00078725 25 12 0.00037788 26 12 0.00017441 27 12 0.00007751 28 12 0.00003322 29 12 0.00001375 30 12 0.00000550 31 12 0.00000213 32 12 0.00000080 33 12 0.00000029 34 12 0.00000010 35 12 0.00000004 36 12 0.00000001

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