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. 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. 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. 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. 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