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this is the first of three presentations ondary widely used uh... probably did in
city functions
and so we're going to talk about the uniform probability distribution which
is probably
simplest
probability density function
off all of that beacause
is nothing but the following
continues and it was set to begin from distribution on an ample a b four years
its probability density function is
uh... info x and then parameters and b
is one oh four pm on the stage for a smaller in quick story could be in zero
otherwise well thats
kinda hard to read basically wanted the uniform distribution does it use its
lines
zero probability to anything outside
twenty interval a_b_
in it
basically as much as this is possiblity continuous signs
the same probability to israel to come
inside the interval
in in order to make sure so that or already means that the function of
course is critical in zero
inst too
make sure it does not vetoed area
office rectangles on this one
of course we must have at the height of suspecting that will ever be months a
and so now if you look at this picture
and you go back to the symbolic deformation you see
that's exactly what symbolic efficient as it gives us zero outside the interval
eight-b_
and it gives us one forty one st
inside the interval
and i really didn't mark herewith with with dot so that that these two points
are anti-*** minus a
because is we have seen for continuous distributions
whether these inequalities are strict or not
really doesn't matter if the probabilities
change
all right now with eight provided it intensified today as this
simple
uh... basically
our interpretation of probability as area already is good enough to compute
probabilities
we don't have any calculus to integrate
constant functions and so we're going to just
talk about rectangles here as we know solve a bunch of problems before we get
there however let's also
uh... ascertained that even though this is a very
convenient insidiously
real density it has really examples mehendi
function
ranges that spreadsheet function rental refer or have prefer to previously
that actually is
uniformly distributed on the interval
from zero to one someplace you can try
take a spreadsheet anti pain ranjith this function
that has
uh... no argument here i got a value of point three oh one seven eight another
spreadsheet a set of such a way that
every time something is changing every time something is a picnic we compute
all the celso
uh... if every country granted it point nine one something at that point nine
seven something ticket point to one something that i get
point four five something in silwan in silwan and so on
and so now if i were
collect these men remains of course i just sort of like collecting outcomes of
coin flips patronage in a spreadsheet
i could just go ahead and cleary is random variable something like two
hundred times inset happy histogram
then basically that histogram is going to
sort of look like our will give us
bars between zero and one res zero zero one is here and all the bars ought to be
a profit synch right they're not going to be
exactly uniform all the way across to police on fluctuations but basically
the experimental outcome will look very much
lightly
uniform distribution so that means
this thing is
implemented heathrow round number generators and the like
and uh... well so now let's take a look at some examples of these examples are
just
designed to
drive home
the point
that probability theory is that we're going to look at one renegade alex that
is uniformly distributed
whatever fruit
from two to five moving to compute
each of the following probabilities probably that exits masonry opened three
well
three ways to as one divided by three is three iraq next week
really want to make sure of their stuff we just go ahead int after all the
distribution of the allies three is here
so that probably that exist there's an article in three years the probability
that exists between two and three eighths accepted this area of this
rectangle incentives
height one-third
times one which is just one for okay
so
eyes keep going with that and if we want to have the property taxes greater
equipment four
well similar situation we look at the pro-nazi density
for is here
which means that this rectangle here
irian tu her presence of probity remember the high just one-third so the
priority of makes great articles and four
it's a probably that exist between four and five
it's still
one-third because sidekicks one-third
uh... if we're looking at the probability exist between three point
five ants ever
well then we have to be a bit careful pecans well three point five is here
and seven eighths somewhere over here already the probability distribution
assigned to probably did you already assigned zero probability
so actually
the probability is completely captured in this rectangle which means
that for this random variable the probability that exist between three
point five and seven is the probability that exist between three point five and
five because politics is greater than five zero
evette ends up being well one point five times
one point five times one-third and that's one half because it's exactly
half the area off the whole thing i couldn't i didn't put the steps and
okay so
now let's keep looking at the random variable that is distributed
uniformly between two and five int
let's look at a different problem let's look at
the problem of computing a value pics
so that the probability that are random variable is gregory or the next
use point seven that's the kind of scenario
even though the probability density functions will be different and may have
to you can be found empirically
that's the kind of things that you
i can't face in design that you want to build something in such a way that
you've got a seventy percent ninety percent
in ninety nine percent chance that something is going to happen within
uh... is or isn't going to happen
briefing the day
certain time that's where a lot of these
abstract computations come in
in twelve pt still
you just have to look at your underlying density function
in this town we'd want and area
to have the right of a point x
in that area is supposed to be
point seven right so we
put a point x here if we want to have that this
totally area here it is supposed to be
point seven
well that means
that
this area here
is
point
free and so that means
one thirty
times
the step to the right from
uh... too
has to be put a point three so one-third timestamp labels pointed out to be
point nine so this ought to be two point nine well all right let's see what the
competition gives us
exists here
insult basically what i have this
point seven divide by one third thursdays name
that works out to you
so thats point seven times three that's two point one
infinity x's five minus two point one
and that would be two point much so
i've worked with the left side which is basically isn't complimentary event if
we want to use
the language that we have also introduced
and either way
i get the right results here end
but you will see that there's no there's a lot of me in his computations
aren't that hard
sulk olympic for a look at for
though a look at u_p_s_ okay so
these competitions are in fact that
what we're trying to drive home
with the uniform distribution certainty is this ideal of probability p area is
also off the value of
drawing
these pictures adroit the pictures isn't mandatory to solve these problems before
more complicated distributions
but i found that it helps
visualize things and it also helps visualize city with people who may have
to go
tomb of the complimentary event which at the latest
kin happened when we were working
with the normal distribution which is actually computationally
rather
complicate but before we get there
the next one that i can plan for years the exponential distribution
so i'll see you there