Question

In: Statistics and Probability

The Probability Integral Transformation Theorem states the following: Let X have continuous cdf FX(x) and define...

The Probability Integral Transformation Theorem states the following: Let X have continuous cdf FX(x) and define the random variable U as U = FX(x). Then U is uniformly distributed on (0,1), i.e., P(U ≤ u) = u, 0 < u < 1. This theorem can be used to generate random variables with an arbitrary continuous distribution function F, if F ^−1 is explicitly known. To illustrate how the method works, you will generate 1,000 random numbers from an Exponential(λ) distribution with rate λ, by following the next 4 steps:

1. Write the c.d.f., FX(x), for X ∼ Exp(3), where 3 is the rate or inverse-scale parameter.

2. Find the inverse-cdf, F ^−1 X , where F ^−1 x (U) = X

3. Generate a uniform random variable U

4. Return X ← F ^−1 X (U)

Expert Solution

We have pdf for Exponential() given as :

Step1 :

We have cdf for Exponential() given as :

For X ∼ Exp(3), where 3 is the rate or inverse-scale parameter.

Step 2:

Finding the inverse-cdf :

Step 3 :

Now, we generate 1000 values from U(0,1)

and find the 1000 random numbers from the Exponential(3) using the relation :

where i = 1,2,3,4,............1000

Step 4 :

So, Generated random numbers are :

Note : Since all 1000 values cannot be pasted here as answer cannot be longer than 65000 characters, First 100 values are pasted.

S.no	U_i	x_i = (1/-3)*ln(1-U_i)
1	0.2617	0.4468
2	0.8781	0.0433
3	0.2308	0.4888
4	0.9478	0.0179
5	0.8306	0.0619
6	0.5039	0.2285
7	0.7492	0.0962
8	0.0273	1.2001
9	0.2206	0.5039
10	0.8487	0.0547
11	0.9693	0.0104
12	0.2071	0.5249
13	0.3503	0.3497
14	0.8838	0.0412
15	0.2960	0.4058
16	0.8159	0.0678
17	0.1033	0.7568
18	0.8499	0.0542
19	0.8337	0.0606
20	0.5602	0.1931
21	0.7996	0.0745
22	0.9669	0.0112
23	0.8827	0.0416
24	0.1610	0.6088
25	0.9121	0.0307
26	0.4105	0.2968
27	0.6326	0.1527
28	0.3978	0.3073
29	0.5574	0.1948
30	0.9640	0.0122
31	0.8339	0.0605
32	0.2181	0.5077
33	0.7942	0.0768
34	0.2486	0.4639
35	0.7307	0.1046
36	0.0666	0.9029
37	0.2590	0.4503
38	0.9416	0.0201
39	0.0323	1.1440
40	0.0927	0.7930
41	0.1711	0.5885
42	0.8854	0.0406
43	0.2094	0.5212
44	0.3892	0.3146
45	0.2379	0.4786
46	0.4974	0.2328
47	0.8916	0.0382
48	0.9228	0.0268
49	0.4920	0.2364
50	0.3830	0.3199
51	0.6516	0.1428
52	0.3083	0.3923
53	0.5915	0.1750
54	0.5614	0.1924
55	0.9539	0.0157
56	0.0858	0.8185
57	0.6132	0.1630
58	0.2081	0.5232
59	0.5519	0.1981
60	0.6563	0.1404
61	0.8731	0.0452
62	0.3216	0.3782
63	0.1126	0.7280
64	0.5306	0.2113
65	0.8016	0.0737
66	0.2200	0.5047
67	0.3518	0.3482
68	0.0501	0.9978
69	0.4080	0.2988
70	0.2744	0.4310
71	0.4599	0.2589
72	0.2342	0.4838
73	0.3637	0.3372
74	0.7841	0.0811
75	0.1000	0.7674
76	0.9104	0.0313
77	0.2352	0.4825
78	0.4010	0.3046
79	0.7597	0.0916
80	0.7845	0.0809
81	0.8998	0.0352
82	0.0098	1.5419
83	0.7373	0.1016
84	0.6061	0.1669
85	0.6733	0.1319
86	0.4678	0.2532
87	0.5447	0.2025
88	0.0865	0.8158
89	0.1825	0.5670
90	0.1144	0.7228
91	0.0067	1.6663
92	0.7242	0.1076
93	0.5125	0.2228
94	0.4523	0.2645
95	0.8397	0.0582
96	0.1951	0.5447
97	0.6610	0.1380
98	0.9107	0.0312
99	0.9891	0.0037
100	0.4639	0.2560

orchestra answered 2 years ago

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