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I've forgotten how to do this, please help!

  • 03-02-2013 12:47pm
    #1
    docmol
    Closed Accounts Posts: 339 ✭✭


    As the title says, too long out of education to remember how to work this one out. Any help appreciated, even just a pointer or the name of a formula.
    If an event has a set chance of happening per unit time (in this case 4% per year cumulative) so 4% year one, 8% year two, 12% year 3 etc. what is the mean time for said event to occur?
    Thanks in advance.


Comments

  • #2
    im invisible
    Closed Accounts Posts: 4,372 ✭✭✭im invisible


    im guessing when the probability reaches 50%, 12 and a half years?


  • #3
    Yakuza
    Registered Users, Registered Users 2 Posts: 5,141 ✭✭✭Yakuza


    It's not quite that simple, I'm afraid.
    You need to think about the chances of surviving (the event not happening) from time unit to time unit as a product of it surviving to the time unit before, then the event happening in that time unit. The expected lifetime (or time to the event occuring) is the sum of these cumulative probabilities.
    t	qx	px	tPx	sum tPx
1	0.04	0.96	0.96	0.96
2	0.08	0.92	0.88	1.84
3	0.12	0.88	0.78	2.62
4	0.16	0.84	0.65	3.27
5	0.20	0.80	0.52	3.80
6	0.24	0.76	0.40	4.19
7	0.28	0.72	0.29	4.48
8	0.32	0.68	0.19	4.67
9	0.36	0.64	0.12	4.80
10	0.40	0.60	0.07	4.87
11	0.44	0.56	0.04	4.91
12	0.48	0.52	0.02	4.94
13	0.52	0.48	0.01	4.95
14	0.56	0.44	0.00	4.95
15	0.60	0.40	0.00	4.95
16	0.64	0.36	0.00	4.95
17	0.68	0.32	0.00	4.95
18	0.72	0.28	0.00	4.95
19	0.76	0.24	0.00	4.95
20	0.80	0.20	0.00	4.95
21	0.84	0.16	0.00	4.95
22	0.88	0.12	0.00	4.95
23	0.92	0.08	0.00	4.95
24	0.96	0.04	0.00	4.95
25	1.00	0.00	0.00	4.95
    (I've rounded the figures for display purposes, but the full values are used for calculating the probabilities).

    The total sum of the tPX column is 4.95 (rounded to 2dp), suggesting an average wait of approx 5 time units before the event occurs. This is also (roughly) when the survival probabilty hits 0.5, a good indication of reasonableness).

    I don't have time to format it neatly, but I hope you can get where I'm coming from. tPx is the previous year's tPx multiplied by this year's px (the probablity of suriving to time unit t is the probability of surving to t-1 multiplied by the probability of surviving time unit t.


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