Question

The probability of winning in a certain state lottery is said to be about 1/9. If it is exactly 1/9, what a random variable represents the distribution of the number of tickets a person must purchase up to and including the first winning ticket? Plot the PMF of this random variable. the distribution of the number of tickets purchased up to and including the second winning ticket can be described by what distribution?

Answer 1

Let X is a random variable shows the number of tickets person need to purchase including winning ticket. Here X has geometric distribution with parameter p = 1/9.

The pmf of X is:

Following table shows the pmf for some values:

X	P(X=x)
1	0.111111111
2	0.098765432
3	0.087791495
4	0.078036885
5	0.06936612
6	0.061658773
7	0.054807798
8	0.048718043
9	0.043304927
10	0.038493268
11	0.034216239
12	0.030414434
13	0.027035053
14	0.024031158
15	0.021361029
16	0.018987582
17	0.01687785
18	0.015002534
19	0.013335585
20	0.011853854
21	0.010536759
22	0.009366008
23	0.00832534
24	0.007400303
25	0.006578047
26	0.005847153
27	0.005197469
28	0.004619972
29	0.004106642
30	0.003650349
31	0.003244754
32	0.002884226
33	0.002563756
34	0.002278895
35	0.002025684
36	0.001800608
37	0.001600541
38	0.001422703
39	0.001264625
40	0.001124111
41	0.00099921
42	0.000888186
43	0.000789499
44	0.000701777
45	0.000623802
46	0.00055449
47	0.00049288
48	0.000438116
49	0.000389436
50	0.000346166

Following is the histogram of the data:

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The distribution of the number of tickets purchased up to and including the second winning ticket can be described by negative binomial distribution with parameters r=2 and p=1/9.