b.glazer wrote:
OK, now I'M the one confused. It's been way too many years since I
had my statistics courses to remember the difference between variance
and co-variance (and giving me the formulas won't educate me
anymore).
Please help me with "plain english" -- the $1 5-play vs the $5
single-line game, for example -- it's my understanding that if I'm
putting the same money at risk per "pull", the multi-line game will
give me less peaks and valleys, with the same long-term expectation
per dollar risked.
Is that correct, or does the higher variance (described above, eg, 19
vs 214 for single vs 100-play), mean I'm wrong?
Re your first comment: It's not necessary to remember the fine points
of the distinction. Just understand, if you wish, that co-variance is
a contributor to the total variance of multi-line play and is a
consequence of the fact that there are multiple hand outcomes on the
draw related to the same dealt hand.
Your "plain english" statement in the second paragraph is exactly on
the nose. The seeming discrepancy that seems to lie in the greater
variance of 100-play is a matter of understanding the units involved
(similar, to an extent, to the fact that 90 cm isn't necessarily
greater than 30 in simply because the number is bigger).
The variance numbers are best understood as measured in betting units,
squared, where the multiplay variance is an aggregate that treats each
individual line played as a unique bet. If the multiplay value were
expressed instead as variance per single whole bet, than the 100-play
value would be 2.14.
So, let's consider the case of $1 single line vs. $.01 100-play.
Measured in alternative units that reflect the entire wager of $5 in
each case, the ratio of respective variance would be 19:2.1 (or about
8x higher for the single line wager).
Variance, like length, can be expressed in various units. For
convention's sake (to ensure that there's no question of the units)
the numbers are expressed in terms of "base" bet units.
···
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I want to add a final comment to this discussion -- it's tempting to
translate the relationship between variance from one game to another
into a direct indication of relative magnitude of the gain/loss
exposure one faces in play of the respective plays. This can't be
done -- relative variance is only a general guide to relative loss risk.
Variance, as a mathematical measure, best describes the range of
outcomes for an event that has a true "normal" distribution (e.g. the
flipping of a penny or roll of a die). Because the measure of payback
from vp hands assigns a different weight to each hand outcome,
short-term variance does not adhere to a "normal" distribution.
However, over a large number of hands (typically referred to as the
"long term"), the distribution takes on a shape that closely
approximates a normal distribution.
The bottom line is this: Shorter to Medium term play results (say,
for example, 100 thousand plays) have a range of outcomes that doesn't
reflect a normal distribution. Therefore, for those purposes,
variance isn't a reliable measure of related loss risk (but generally
still good for a gut sense of relative risk). Over the long term,
variance is much more reliable.
One example to illustrate where variance is entirely misleading where
it comes to session bankroll risk: pick'em has a MUCH lower variance
vs. JB (or any other standard vp game). However, experienced pick'em
players know that you better be packing a good number of extra bucks
to have the same survivability in a session as you might playing JB at
the same denomination.
- Harry