2. Definition of 'Law Of Large Numbers'
A principle of probability and statistics which states
that as a sample size grows, its mean will get closer
and closer to the average of the whole population.
The law of large numbers in the financial context has a
different connotation, which is that a large entity
which is growing rapidly cannot maintain that growth
pace forever.The biggest of the blue chips, with
market values in the hundreds of billions, are
frequently cited as examples of this phenomenon.
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3. Central LimitTheorem: Average losses from a random
sample of n exposure units will follow a normal
distribution.
Regardless of the population distribution, the
distribution of sample means will approach the
normal distribution as the sample size increases.
The standard error of the sample mean distribution
declines as the sample size increases.
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4. 4
, -For example a single roll of a six sided die produces
, , , , , ,one of the numbers 1 2 3 4 5 or 6 each with equal
probability. ,Therefore the expected value of a single
die roll is
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5. According to the law of large numbers, if a
large number of six-sided die are rolled, the
average of their values (sometimes called the
sample mean) is likely to be close to 3.5, with
the precision increasing as more dice are
rolled.
.
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6. It follows from the law of large numbers that
the empirical probability of success in a series
of Bernoulli trials will converge to the
theoretical probability. For a
Bernoulli random variable, the expected value
is the theoretical probability of success, and
the average of n such variables (assuming they
are independent and identically distributed
(i.i.d.)) is precisely the relative frequency .
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7. For example, a fair coin toss is a Bernoulli
trial. When a fair coin is flipped once, the
theoretical probability that the outcome will
be heads is equal to 1/2.Therefore, according
to the law of large numbers, the proportion of
heads in a "large" number of coin flips "should
be" roughly 1/2. In particular, the proportion
of heads after n flips will almost surely
converge to 1/2 as n approaches infinity.
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8. Though the proportion of heads (and tails)
approaches 1/2, almost surely the absolute
(nominal) difference in the number of heads and
tails will become large as the number of flips
becomes large.That is, the probability that the
absolute difference is a small number,
approaches zero as the number of flips becomes
large. Also, almost surely the ratio of the
absolute difference to the number of flips will
approach zero. Intuitively, expected absolute
difference grows, but at a slower rate than the
number of flips, as the number of flips grows.
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9. When an insurer increases the size of the
sample of insureds:
Underwriting risk increases, because more insured
units could suffer a loss.
But, underwriting risk does not increase
proportionately. It increases by the square root of
the increase in the sample size.
There is “safety in numbers” for insurers!
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10. When an insurer increases the size of the
sample of insureds:
Underwriting risk increases, because more insured
units could suffer a loss.
But, underwriting risk does not increase
proportionately. It increases by the square root of
the increase in the sample size.
There is “safety in numbers” for insurers!
Prepared by :Reymart Bargamento
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