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December 28[edit]

Oldest Person in the World[edit]

At what average interval should we expect the oldest person in the world to die? Let's assume that age is a normal distribution and that the population of the world is roughly constant. --Auximines 11:40, 28 December 2006 (UTC)[reply]

Let's not; those assumptions are decidedly counterfactual! Look at the world population distribution and the U.S. population distribution to see that population percentage roughly decreases with age. (No surprise.) And the world population has been steadily increasing for decades, with declines in European countries more than offset by increases elsewhere. --KSmrq^T 12:22, 28 December 2006 (UTC)[reply]

After edit conflict, - tend to agree with KSrq, calc here as they might be a nice example of normal dist but no bais on reality.

First off the normal distribution is probably a poor model for life expetancy, the actual distribution is likely to be more complex. Lets make a few assumptions, say the Life expectancy or mean life span is 70 (i.e. three score years and 10), and the standard deviation is 10 years and there are 5,000,000,000 people on the planet.

From normal distribution we have

68.26894921371% of the area under the curve is within one standard deviation of the mean.
95.44997361036% of the area is within two standard deviations.
99.73002039367% of the area is within three standard deviations.
99.99366575163% of the area is within four standard deviations.
99.99994266969% of the area is within five standard deviations.
99.99999980268% of the area is within six standard deviations.
99.99999999974% of the area is within seven standard deviations.

Or

15.865525393145% is greater than 1sd. I.e. greater than 80
2.27501319482% is greater than 2sd. Greater than 90
0.134989803165% is greater than 3sd. Greater than 100
0.003167124185% is greater than 4 sd. Greater than 110
0.00005733031% is greater than 5 sd. Greater than 120
0.00000009866% is greater than 6 sd. Greater than 130

Multiply these by the population of the earth (5,000,000,000) gives

6749490.15 greater than 100
158356.2 greater than 110
2866.5, greater than 120
4.933, greater than 130

Not bad considering the wild asumptions at the start. This are very silly using normal distribution for young ages as you often get high infant mortality. --Salix alba (talk) 12:57, 28 December 2006 (UTC)[reply]

I don't think I phrased the question very well. I was wondering how often we should expect the record for Oldest Person in the World to change hands. It's currently Emiliano Mercado del Toro and has been since 11th December. Prior to him it was Elizabeth Bolden who held the record for about 3 months. --Auximines 17:44, 28 December 2006 (UTC)[reply]

Let us first make the simplest possible assumption, namely, that the probability to die for a given person during a certain (very short) interval of time is simply constant; that is, depends neither on the age of that person nor on any other factor. Then, life-span distribution would be Poisson: P(t)dt = exp(-t/t₀)dt/t₀, where t₀ is average life-span. It is then trivial to see that the distribution of the "reign intervals" for the longest-living individuals would be the same Poisson distribution. Indeed, once you are the longest-living individual you still have the same chance to die at any given moment, so your reign duration is given by the same probability distribution. Now, this model is of course badly flawed: as the person ages, his/her health deteriorates deterministically, to the point that the probability of death per unit time increases substantially with age. The stronger it increases, the shorter is the expected average reign of the longest-living individual. So, it is very model-dependent. I do not know which model is used in actuary calculations, but perhaps the prices of health insurance per year, as a function of age, could yield a decent estimate. Cheers, Dr_Dima

The distribution that Dr_Dima mentioned is the exponential distribution, not the Poisson distribution. In any case, any method for evaluating this interval will require some hard data, which can be somewhat unreliable since there are few people who reach the ages of our interest. Therefore, I believe that a much better approach than trying to come up with all sorts of fancy models for the problem, is to simply calculate the average interval for this event in the past, that is, divide the length of time a record was kept by the number of times the event occured in that period of time. I got a figure of roughly a year and 5 months. -- Meni Rosenfeld (talk) 19:11, 28 December 2006 (UTC)[reply]

As a further illustration of the sensitivity of the dependence on the distribution mentioned by Dr_Dima, consider another very simple model, in which every individual has exactly the same lifespan t₀: a one-point distribution. Then everyone becomes eventually, for a fleeting instant, the oldest living person before passing on the baton. Assuming a stable world population of size N, the average duration is t₀/N. Compare this to the value t₀ for the negative exponential distribution. --Lambiam ^Talk 21:18, 28 December 2006 (UTC)[reply]

Dima's model is, however, somewhat reasonable, in the sense that the exact time of a person's death is sufficiently unpredictable that dying can be locally approximated as a Poisson process: that is, your chance of dying tomorrow is non-zero, and is approximately the same as your chance of dying the day after tomorrow, or on any given day a week, a month, or even a year from now, assuming you survive until them. Assuming that most people who achieve the title of oldest person alive only hold it for a short enough time for this approximation to be globally valid — which is to say, for only a small fraction of the standard deviation of the human lifespan — then it makes sense to speak of the "average mortality rate of the oldest person in the world", which simply equals the average number of times this title changes hands in a unit span of time. —Ilmari Karonen (talk) 03:36, 4 January 2006 (UTC)[reply]

i have trouble[edit]

an eight-year old boy named nathan wants to compare his age with that of his mother and grandfather. His mother is exacly four times his age and his grandfather is exacly seven times his age. How old is nathan's mother? how old is his grandfather?

i dont understand this part.

1. underline the key pieces of information contained in the problem statment.

2. can u rewrite the problem statement in the firm of an equation? if so, what is required for you to write it?

3. can the mother's age be found using the same problem solving strategy as the one use to find the grandfathee's age?explain

4. solve the equation. how old is nathan's mother? how old is his grandfather?

5. one year from now, will the same relationship betwenn the ages of various family members could have been found wothout using equations and variables. describe a situation in which it would be necessary to employ the more formal mathemathical techniques of variables and equations rather than simple reasoning.

so can u please help me to understand the problem and how to solve it. —The preceding unsigned comment was added by 71.114.83.58 (talk • contribs) 19:06, December 28, 2006 (UTC).

Wow! This is a tough challenge. The problem is so simple and so clearly stated, guiding you through solution, that we can hardly say anything without essentially doing your work for you. (As clearly stated at the top of this page, you can ask for help with homework, but we will not do it for you.)

Let's step back and ask what the problem is about. Are we interested in how Nathan feels about his mother? No. Are we interested in his grandfather's eyesight? No. We must focus on the mathematical content. We look for numbers; three are given. We look for relationships among those numbers; two are given. These are examples of what is given; we also must know what is asked.

The numbered questions are meant to help you with each step of understanding and solving this specific problem. They teach you a strategy. You may also wish to read George Pólya's How to Solve It. He gives suggestions that are similar to those here, and that may help even when the problem is much harder. --KSmrq^T 21:07, 28 December 2006 (UTC)[reply]

......................

Is there any part of the problem that you do understand? Could you point out where it is that you get lost?

Suppose someone tells you: "I have solved the problem. Nathan's mom is 40, and his grandpaw is 72." What do you think? Can this be right? --Lambiam ^Talk 21:31, 28 December 2006 (UTC)[reply]

If Nathan was 7 and his mom was 3 times as old, you would multiply 7 by 3 to get 21 as the mother's age. Now try it using the numbers they gave you. StuRat 01:17, 29 December 2006 (UTC)[reply]