馃攳
Geometric distribution mean and standard deviation | AP Statistics | Khan Academy - YouTube
Channel: Khan Academy
[0]
so let's say we're going to play a game
[2]
where on each person's turn they're
[4]
going to keep rolling this fair
[6]
six-sided die until we get a one
[11]
and we just want to see how many roles
[13]
does it take so let's say we define some
[16]
random variable let's call it x and
[18]
let's call it the number
[20]
of rolls
[22]
until
[24]
we
[25]
get
[27]
a
[28]
one
[29]
so what's the probability
[31]
that
[32]
x is equal to 1. pause this video and
[36]
think about it
[37]
all right the probability that x is
[39]
equal to 1 means that it only takes us
[41]
one roll to get a 1. well that's going
[43]
to be a one sixth probability
[47]
well what's the probability
[49]
that x is equal to two
[52]
well that means that we on the first
[54]
roll we get something other than a one
[56]
so that is going to be 5 6
[59]
and then on the second roll we get a 1
[62]
so that has a 1 6
[65]
probability
[66]
and we could keep going what's the
[67]
probability that x is equal to 3 pause
[71]
the video and think about that
[73]
well that means we miss on the first two
[76]
so we have a 5 6 chance of getting
[77]
something other than a 1 on the first 2
[79]
rolls so we could say that's 5 6
[82]
times 5 6 or we could write 5 6 squared
[85]
and then on the third roll we have the 1
[88]
and 6 chance of getting the 1.
[90]
so
[90]
times 1 6. and i think you see a pattern
[93]
here and you might recognize what type
[95]
of random variable this is this is a
[98]
geometric variable now how do we know
[101]
that well each trial or each role is
[104]
either a success or a failure every time
[107]
we roll we either get a one or we don't
[110]
we have the same probability of success
[112]
of rolling a one each trial these are
[114]
independent trials
[116]
and that there's no set number of trials
[119]
it could take us an arbitrary number of
[121]
trials to get the first success so
[122]
that's what tells us that we're dealing
[124]
with the geometric random variable
[127]
now one question is is what is going to
[130]
be the mean of this geometric random
[133]
variable well we proved another video
[135]
where we talk about the expected value
[137]
of a geometric random variable we're
[138]
really talking about the mean of a
[140]
geometric random variable
[142]
and it is a little bit intuitive if you
[144]
were to just guess what is the mean of a
[146]
geometric random variable where the
[147]
chance of success on each roll is 1 6
[149]
you might say well maybe on average it
[151]
takes you about six tries and you would
[153]
be correct the mean of a geometric
[155]
random variable is one over the
[158]
probability of success on each trial so
[161]
in this situation
[163]
the mean
[165]
is going to be one over the probability
[167]
of success in each trial is one over six
[170]
so it's equal to six so one way to think
[173]
about it is on average you would have
[175]
six trials until you get a one now
[178]
another question is what's a measure of
[181]
the spread of a geometric random
[183]
variable and we don't prove this in
[185]
another video maybe i'll do it
[186]
eventually that the standard deviation
[190]
of a geometric random variable is the
[193]
mean times the square root of 1 minus p
[196]
or you could just write this as a square
[197]
root of 1 minus p
[200]
over
[201]
p
[202]
now in this situation what would this be
[204]
well the standard deviation
[206]
of this random variable this geometric
[208]
random variable it's going to be the
[210]
square root of
[212]
1
[213]
minus 1 6
[215]
all of that over
[216]
1 6.
[218]
so this is going to be equal to the
[220]
square root of 5 6
[223]
over 1 6 which is equal to 6 times the
[227]
square root of 5 6 and this is going to
[230]
be approximately equal to 5
[233]
divided by 6 is equal to that we'll take
[236]
the square root of that and then
[238]
multiply that times 6
[240]
gets us to about 5.5
[243]
so approximately equal to 5.5
[247]
and what's interesting about a geometric
[249]
random variable obviously the lowest
[250]
value here in this case is 1 2 three can
[253]
go higher and higher but it can go
[254]
arbitrary you could get really unlucky
[256]
and it might take you a thousand rolls
[258]
in order to get that one it could take
[259]
you a million rolls very low probability
[261]
but it could take you a million rolls in
[263]
order to get that one and so another
[265]
thing to realize about a geometric rand
[268]
variables distribution it tends to look
[271]
something
[273]
like this
[274]
where the mean
[277]
might be over here and so you have a
[279]
very long tail to the right of your mean
[281]
and this is classic
[283]
right skew
[284]
and so all geometric random variables
[286]
distributions are right skewed they have
[289]
a long tail of values and infinitely
[291]
long tail of values they can take to the
[293]
right
[294]
now one last question instead of dealing
[295]
with a six sided die what would be the
[297]
situation if we were dealing with a 12
[300]
sided die what would then be the mean of
[302]
our random variable and what would be
[304]
the standard deviation of our random
[306]
variable pause this video and think
[307]
about that
[309]
well the mean would be 1 over 1 12
[312]
because you have a probability of 1 12
[314]
every time of getting a 1 we're assuming
[316]
we're playing the same game now with the
[317]
12 sided die so 1 over 1 12 would be 12.
[321]
so on average it would take 12 rolls to
[324]
get that first one and then our standard
[327]
deviation is going to be essentially
[329]
this times the square root of 1 minus 1
[333]
12. okay let me write it this way it's 1
[335]
minus 1 12 over 1 over 12 which is the
[339]
same thing as 12 times the square root
[342]
of 11 12.
[344]
11 divided by 12 is equal to
[348]
take the square root and then multiply
[351]
that times 12 and you get about 11.5
[355]
11.5
[358]
and so you can see with a 12 sided die
[360]
it has the same pattern where you have
[362]
your mean of your random variable and
[365]
then you have a standard deviation
[367]
that goes
[369]
a reasonable bit on either side of the
[370]
mean it's almost equal to the mean in
[373]
actually in both situations it's a
[375]
little bit lower than the mean but then
[377]
there's many many many values that go
[379]
far to the right of your mean and so you
[381]
have this classical right skew for a
[383]
geometric random variable
Most Recent Videos:
You can go back to the homepage right here: Homepage