“Doctors say he’s got a 50/50 chance of living… though there’s only a 10% chance of that.”

I’ve lately had occasion to contemplate my own mortality. How long should I expect to live? The most recent life table published by the Centers for Disease Control (see the reference at the end of this post) indicates an expected lifespan of 76.5 years for a male. This is based on a model of age at death as a random variable with the probability density shown in the following figure.

The expected lifespan of 76.5 years is (using the red curve for males). In other words, if we observed a large number of hypothetical (male) infants born in the reference period 2014– and they continued to experience 2014 mortality rates throughout their lifetimes– then their ages at death would follow the above distribution, with an average of 76.5 years.

However, I have more information now: I have already survived roughly four decades of life. So it makes sense to ask, what is my conditional expected age at death, given that I have already survived to, say, age 40? In other words, what is ?

This value is 78.8 years; I can expect to live to a greater age now than I thought I would when I was first born. The following figure shows this conditional expected age at death , as well as the corresponding expected additional lifespan , as a function of current age .

For another example, suppose that I survive to age 70. Instead of expecting just another 6.5 years, my expected additional lifespan has jumped to 14.5 years.

Which brings us to the interesting observation motivating this post: suppose instead that I die at age 70. I will have missed out on an additional 14.5 years of life on average, compared to the rest of the septuagenarians around me. Put another way, at the moment of my death, I perceive that I am dying 14.5 years earlier than expected.

But this perceived “loss” always occurs, no matter when we die! (In terms of the above figure, the expected value is always positive.) We can average this effect over the entire population, and find that on average males die 12.2 years earlier than expected, and females die 10.8 years earlier than expected.

Reference:

Arias, E., United States Life Tables 2014, National Vital Statistics Reports, 66(4) August 2017 [PDF]

Following are the probabilities for the United States 2014 period life table used in this post, derived from the NVSR data in the above reference, extended to maximum age 120 using the methodology described in the technical notes.

Age P(all) P(male) P(female) =========================================================== 0 0.005831 0.006325 0.005313 1 0.000367843 0.000391508 0.000343167 2 0.000246463 0.000276133 0.000216767 3 0.000182814 0.000206546 0.000157072 4 0.000156953 0.000183668 0.000129216 5 0.000141037 0.000160804 0.000120255 6 0.000125127 0.000142914 0.000106328 7 0.000112203 0.000128008 0.0000963806 8 0.000100276 0.000112117 0.0000884231 9 0.0000913317 0.0000992073 0.0000834481 10 0.0000883454 0.0000932456 0.0000824477 11 0.0000952854 0.000103156 0.0000874072 12 0.000119095 0.000137857 0.000101304 13 0.000164729 0.000203286 0.000124134 14 0.000227209 0.000294457 0.000155893 15 0.000293617 0.000391501 0.000190617 16 0.000362946 0.000491412 0.000227306 17 0.000442117 0.000609009 0.000265957 18 0.000529115 0.000743227 0.000302594 19 0.000616971 0.000881116 0.000338207 20 0.00070566 0.00101964 0.000373784 21 0.000786255 0.00114394 0.000408332 22 0.000847887 0.0012343 0.000438875 23 0.000886654 0.00128494 0.000465417 24 0.000909534 0.00130883 0.000490933 25 0.000929392 0.0013228 0.0005174 26 0.000952147 0.00134161 0.000545804 27 0.000977786 0.00136426 0.00057515 28 0.0010063 0.00139366 0.000605432 29 0.00103864 0.00142878 0.00063566 30 0.00107285 0.0014657 0.000668788 31 0.00110792 0.00150147 0.000705793 32 0.00114483 0.0015361 0.000745674 33 0.00118454 0.00156959 0.000792356 34 0.00122898 0.00160582 0.000845811 35 0.00128495 0.00165443 0.000908956 36 0.00135141 0.00171632 0.000981746 37 0.00142538 0.00178654 0.00106021 38 0.00150387 0.0018631 0.00114136 39 0.00158781 0.00194786 0.00122516 40 0.00168489 0.00204935 0.00131843 41 0.00179886 0.00217314 0.00142304 42 0.00192761 0.00231898 0.00153401 43 0.00207581 0.00249327 0.00165615 44 0.00224899 0.00270038 0.00179419 45 0.00243725 0.00292846 0.00194116 46 0.00265083 0.0031884 0.00210855 47 0.0029103 0.00350311 0.00231349 48 0.00321588 0.00387243 0.0025555 49 0.0035468 0.00427362 0.00281771 50 0.00387592 0.00467212 0.00307957 51 0.00420287 0.00507013 0.00333606 52 0.00454693 0.00549737 0.00359937 53 0.00492128 0.00597165 0.0038758 54 0.00532664 0.00649003 0.00417147 55 0.00575619 0.00703581 0.00448667 56 0.00619215 0.00758275 0.00481225 57 0.00662626 0.00813029 0.00513737 58 0.00705499 0.00866988 0.00545972 59 0.00748745 0.00921066 0.00578812 60 0.00794918 0.00978879 0.0061402 61 0.0084469 0.010399 0.00653116 62 0.0089597 0.010994 0.00696556 63 0.00947691 0.0115436 0.00744933 64 0.0100035 0.0120603 0.00797895 65 0.0105466 0.0125691 0.00854801 66 0.0111425 0.0131347 0.00916566 67 0.0118165 0.0137895 0.00985298 68 0.0126025 0.0145881 0.010625 69 0.0135386 0.0155721 0.0115183 70 0.014622 0.016711 0.0125556 71 0.0157853 0.0179169 0.0136863 72 0.0169733 0.0191484 0.0148415 73 0.0181664 0.0203416 0.0160437 74 0.0193544 0.0214907 0.0172763 75 0.0205581 0.0226235 0.0185559 76 0.0219039 0.023887 0.0199909 77 0.0233782 0.0252875 0.0215559 78 0.0249405 0.0266573 0.0233399 79 0.0266659 0.0281283 0.0253501 80 0.0283006 0.0295587 0.0272207 81 0.0298041 0.0307938 0.0290306 82 0.0311707 0.0318902 0.0307088 83 0.0326118 0.0329808 0.0325375 84 0.0338734 0.0336728 0.0344093 85 0.0348103 0.0342521 0.0357896 86 0.0356915 0.0345144 0.0373244 87 0.036144 0.0342741 0.0384714 88 0.0361034 0.0334914 0.0391388 89 0.0355212 0.0321521 0.0392438 90 0.0343716 0.0302738 0.0387212 91 0.0326583 0.0279093 0.0375332 92 0.0304192 0.0251463 0.0356786 93 0.0277276 0.0221028 0.0331989 94 0.02469 0.0189177 0.030181 95 0.0214386 0.0157381 0.0267542 96 0.0181203 0.0127037 0.0230805 97 0.0148823 0.00993297 0.0193396 98 0.011857 0.00751144 0.0157101 99 0.00914934 0.00548606 0.0123497 100 0.00682791 0.0038652 0.00937893 101 0.0049216 0.00262443 0.0068711 102 0.00342273 0.00171608 0.00484973 103 0.00229461 0.00108015 0.00329442 104 0.00148199 0.000654343 0.00215221 105 0.000921789 0.000381551 0.00135156 106 0.000552114 0.000214241 0.000815772 107 0.00031851 0.000115918 0.00047333 108 0.000177059 0.0000604924 0.000264139 109 0.0000949139 0.0000304833 0.000141878 110 0.0000491106 0.0000148533 0.0000734294 111 0.0000245565 0.00000700891 0.0000366663 112 0.0000118822 0.00000320819 0.0000176914 113 0.00000557214 0.00000142697 0.00000826206 114 0.00000253659 0.000000617876 0.00000374136 115 0.00000112287 0.000000260924 0.00000164594 116 0.0000004842 0.00000010766 0.00000070483 117 0.000000203758 0.0000000434809 0.000000294373 118 0.0000000838243 0.0000000172192 0.000000120143 119 0.0000000337717 0.0000000066978 0.0000000480077 120 0.0000000216956 0.00000000409582 0.0000000304297