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normal distributions are is the third and last of three families are
frequently use distribution stepper investigating in this section
and they are arguably
than most frequently used family off of probability distributions because on one
hand
uh... quite a few sarcastic phenomenon of a mess but in an example many
scientific measurements
are nobody distributed or at least
there is soon to be normally distributed
uh... no distributions and also arise through that very important central
limit theorem which we will
discuss later on in these presentations
and off that basically means that
because essentially maturing talks about average is an average are being taken
quite a few times that
normal distributions will be permeate
all of statistics
however we also need to remind ourselves
not everything is normally distributed is often a standard assumption in
statistics
that certain phenomenon is normally distributed
but technically that also has to be tested because it need not be true
but of those kinds of tests in concentrations are
beyond hope we're going to be
okay so what is the normal distribution well
a continuous random variable which we will denote in
set new sigma normal mu sigma
is said to have a normal distribution with the perimeters and you
in sigma
if you cut the following probability density function it's difficult muse
excitement signal x
eyes one overstate my route
hi each of the negative x months new quantities quit
developer to sigma squared
which is uh... that kind of function
that we've team to you quite a few of the earlier presentations
so that we don't need to do
too many of the computations here at least not
worry about it too much
uh... well
new can be any real number and uh... sigma is any positive real number
indict something that really uh... investigate alternator presentation
instant vocabulary yet
the parameter in u_s_ called the expected value off the round table
in the parameter sigma is a standard deviation of surrounding variable
those are just means right now but we will bring some work
meaning to them in a higher personal touch
okay so
one of the things that needed to know your presentation actually a sweet
sketched a graph of this family a functions and we ended up
with this picture we're not going to
go through all the details here but basically
in the presentation on graphing functions with parameters we saw
this family of functions basically looks like a
uh... every one of these functions looks like a bell shaped curve
the center of the bill is that exit polls and you
in the width of the delegates
uh... is is
somewhat to determine by the sigma that is neither meaning the fact that
mu money segment you classic martin section points
front seat of a rebel calc front you know there was a different station
i thought actually is not the reason by segment is is that important the reason
by segment importance because it
it covers them
weeks
uh... the curtain
and uh... welcome kit we also see at least some of the shape from here well
yes we can
because negative x clear it is something
that and gets more and more negative theory rapidly as we get away from the
center
and that would mean that features the negative x squared has to
geeky rather rapidly which is what it does
and constant if we had any
large signal wasn't the case uh... wonderfully small sigma
then decay as fast reaching us
elected to do with the with parameter
that is something that we can also book at in a more dynamical fashion
manual
well you were just uh... gives us the center and so if you if you track
the value of new is this distribution shifts around
you realize that new religious just
places the center of this bell curve
uh... at wherever the value of munies
without also makes sense uh... in terms of what we have learned about
shifting stretching and reflecting because
if segments constant and this is a function of excellence u_n_
muris essentially just air right ships off the craft by exactly you units
uh... if we're looking at sigma
well the effect of segment yes
that's as i mentioned large segment means that
perfect black and white small stigma means it gets to let me know
either way i remember that for
for probability density functions courtney riyadh kore
must be one
impedes scaling factor is with also mention agrees presentation of the
scaling factor that
looks rather strange
makes sure that the total of the area that is on the distribution is always
equal one
n_p_r_'s anne
large segment means its
small segments
it against narrow
all k
so
next thing we need to think about is how do we compute for a vote is for a moment
to strategic randy ripples
for
on
for uniform distributions we just
looked at rectangles
for exponential distributions we just computed integrals
is we have also seen earlier presentations or at least as was
mentioned
uh... into the native x squared is something that doesn't have a nice in
contributed so
we need to think of something here again the first step is
that we're going to take
archery knowledge distributed and bearable with me and you would be ent
uh...
standard deviation signal
and we're going to just look at the probity that such a random variable
takes values between a and b_ in of course
that's just the integral to the density function
from h_i_v_ and
problematic part here is that we've got parameters incite the integral
let's see if we can get rid of those
the first step is just very simple after we keep the fact route front
in crete
rewrite the exponent and realize that excellence mu quantities great divide by
segments where it is of course
aches minus new
divided by signup
quantity
squared
to get rid of of
this term here we can just do a
very simple
linear substitution
z_ equals excellence mute about passing out
that would give us d_c_ d_x_ whatever sigma
content d_x_ bs
sigma deepti
the xmen with me over signup
becomes a seat
the factor out front states the site
inst
the boundaries of course here's one of the
situations where we need to remember that would be
i do substitution was a definite integral because transform the
boundaries uncertainty welding x's any
then see is a minus me over sigma
it's any
z as being mostly oversee
and so that means
now that we
can also
cancel the signals here
we realize that the probability that an artery normal random variable
is between a and p
cindi expressed as an integral
off eight
density function that doesn't have any parameters
and the parameters have been pushed into the boundaries
which i mean it doesn't get a threat of the parameters it shouldn't because all
of these are different
distribution certainty
parameters in the ballary zara holbrooke easier to deal with
then parameters inside the endocrine pretty which is why
this function as a special meaning
which means that
uh... if we have a continuous wonderful c
it has i think it's going to be called alasdair knowledge of the diversity of
the central distribution
if it only if its probability density functions this function from the
previous panel
zero one newest zero segments one policy which is went over to the client is a
negative
the sea screwed over too
in the presentations on
uh... numerical analysis we have also
gotten as close as we can't seem to be able to approve
that indeed
this function is a privilege pinsky functions at defeating the purpose
function over the realign
use one
okay so now
we can say the following if we've got in normally distributed random variables
parameters and human segment
random variables even is obtained by subtracting you and dividing by signal
i think is a standard normal distribution in the probability that our
original randhir logistically any
in bs the credibility is at work
normally distributed enviable is doing a month's new over sigma
imbue months you oversee
uh...
z
rule of thumb essentially that whenever we work with they normally distributed
and variable
we've been in our computations typically subtracted u
identified by the signal that is something that will stay with us
uh... in terms of proving this result what we already have shown
the equation for the probabilities on
uh... two panels they don't
uh... we still would take if you have to establish that this kind of variables e
equals
uh... normal and their demise data but the stigma has a standard normal
distribution but that's not parties there because
the probability
that
random variable
minus you divide it by segments between a month's new over cement b_-minus
new signal that's the problem is at the
actual random variable is between
alien bien that he is
integral off the standard or distribution
from one bound to the other bun and these are exact the balanced which means
if i wanted to make the out of a more complicated now say
seen smaller iqbal this article be and i would get c
indeed right here so
this transformation
returns any
an archery norm and they're going to a standard
normally distributed threatened
okay let's take a look at some examples and
basically we're going to talk about stealing here for a while
uh... and so let's consider any
n invented
steel manufacture
who produces steel rods in the mean hardness off uh... sixty eight
rockwell two units hardness of steve this mission
in rockwell units which are dimensionless and becoming three favors
recall one rockwell to rockwell three
brought well too mister
uh... intermediate hardness if you will in values are typically around the
seventies somewhere
so
although this is a made-up problems of the arguments are
something that is that it is reasonable
it's something that i mean engineer wouldn't ask me out
out of the room with
and has also said that the hardness authorizes nobody distributed with the
about mean sixty eight
properly guinness trouble too
instead of deviation said michael still not sure
how realistic such a
large standard deviation is but
we can now start doing probability was that because basically what you have to
consider
uh... when you are manufacturing something especially when you're
manufacturing
raw materials is that there is some kind of a star
and so
what we want to have what we want to find out here for example is what the
probability is that a randomly chosen across hardness
if at most
seventy
uh... inch
well that that's just
because accommodation of our gratitude to you and a half at the heart it's at
least something cause hardness
for steve is a good property as long as it doesn't
actors and adversely effect flexibility instead
opaque so
the pro our grand airbus the hardness and so we want to know the probability
that the hardness assessment program seventy
well the hardest is nobody distributed with mean sixty eight and standard
deviation
to so
we subtracted nina divide by the standard deviation and typically the way
we handle we will have a package that
when the random variable side we release attract the symbolic quantities
whereas on the numerical slack we subtract the numbers
then he knows that
normal right there but my guess thats mean divide by sigma
that use any
standard on a rented a room which is typically you know if i see
and while the other part is insulting the easy cost seventeen months sixty
eight of apec to as well
okay so we need to figure out the probability that's the useless in repose
in one and threatening to clinton's an integral
for which we don't have been nice integer attitude and so typically
people look at these numbers
normal distribution tables
such as this one
cute extended normal distribution function and we will also talk about
humanity
distribution function thing the next presentation
this function records the area to the left off the bat he makes that you put
into the function so it's the end of may to phoenix
off the cerebral distribution
in so that means
entry here should give us the area to the left of one
which is point eight four one three that maybe a little small to see
in the recording but we're going to just
go back here and we realized okay it's about point eight four one three so it's
about and
eighty-four percent chance that the hardness of the randomly chosen draw
is at most seventy rockwell to use
the rule of thumb that will stay with us it's *** tractor you
when we divide by the signup
and we need to remember
that the table gives the area to the left off the point
mainly and the picture that we would then want to grow as always set
wherever our x's
the table gives us the area that is to limit thinks that that's what
that's what you're gives us
i'm not sure if the states quite accurate but
as a generic picture for normal distribution district
okay so now it's just staying with this and see what other kinds of promoted as
we can compete because also
determined the probability that a rally chosen wright's *** sixteen seventy
one
that the story sustained we just have to say that the committee quality probably
departments craters and sent me one
we subtract new but if i'd say thereby i said no we do it symbolically for that
around and it'll be used in numbers
on zach's site with numbers we end up with the probabilities that a
which then all around the vehicle is greater than three yassin
here we have a little bit of a problem because the table doesn't give us a pro
you that the is greater than three house
it only gives us
the complimentary probability which is exactly this phiri right and so one of
the things that were used to to work with complimentary probabilities work
with the fundamental theorem of calculus
because we realize that in order to get the area to the right overvalued
we might as well
work out the area to the left off that that gets attracted from what so
the probability that z is greater than three has
is one minus the productivity is less than or equal in three house
that probability of course we can figure out because
three houses one point five
in so we end up with a provocative point nine three three two
that we are today after three years
and when you do this attraction that means you've got point or six sixty
eight priority of point o six six eight
to be
odd to have a hard as it is greater than seventy one
alright we keeps saying with this and are now we want to know the productivity
randomly chosen cross hartness
is at most sixty cent
okay gravity of the hardness
listen or equal to sixty-seven subtract new divide by signal we end up with a
priority that a state of rama distributed random variable
is this eric was a negative one-half
again we have a problem
the table in fact a table only gives us values
foresee being
greater
than zero
now there are there are two ways around that antonin
quite a few books you also have a table
that includes negative value site
chose not to do that here because we can also use of the liberal symmetry
do some more graphical reasoning play a little bit more with the meaning
off the integral as in the area and so if i want to know
the probability that he has this article that one
made if one half twelve and im
looking for a tail probability
to the lift off the light access
but decries
the scared normal distribution asymmetric across the white axis
approval of the
that led to the left of the native exits the same s
that we are to the right
off the corresponding positive number
here we have to be careful because my starts in may to fix in the positive
number has a negative sign right
but so basically that means that this probabilities of probity oh c_b_n_ gray
rickles and one half
and wynona compute those because we can just go ahead and computer
one minus
the probability of the compliment
and uh... well the probability
that he is smaller equipment point five years
point six nine one five here
and so if we
go back to the ticket one minus point six nine one five in some of the results
of these
but the probability that the harness is innocent
or equal to sixty seventy s
point three zero
igf-i
okay what else can we do
well we can figure out the probability of that
a randomly chosen prosperous as between two numbers if we have a certain range
say somebody
order stand and says look it's it's supposed to be
at least heavy storms at sixty three and most arms at seventy two which
may be determined by considerations that if you make steel too hard actually gets
brutal
it with this process that made
with whatever crosses being used he writes a fictitious
setup it it may well be back
too much harder sweetie
introduces his comfortable nest
well then we have to set up the probabilities that our
randhir atlas between sixty six in september
uh... which
we do the same things before we subtract new divide by signal is just that now we
have to do it
both sides we have to do it for two numbers well here we get negative wanted
you to get close to
so this is the probabilities that any
standard normal random variable
besides the years between
native one into and of course we know now if we
look at that if we're interested in the probability
that day
mixed and already devastating any n_b_a_ were interested in this area
which means for interested in the area that were to the left b
the repairs to the left of the
which is the property that's useful too
minus the area that we're that is to the next update which is a productivity is
sort of any one
and uh... well then we know that this is part of the probability of cease more
than two
minus one minus the probability is it's the is smaller than one right
same
transformation as we did
in the previous example
in dot will then be just clean up the native signs and we've looked up stuff
in the table enough now
so we get that
probity of the smaller than to his point nine seven seven too
the probability of the slight oneness point eight four one three
if we had those two numbers and subtract one of the pit
beget point eight one eight font
so we've got an eighty one percent chance that within this range which
presumably could be the quick
all right i think this should now be
final problem even in this setup of examples were aboard still with the same
steel manufacture and
basically in manufacturing processes give you've got to have ways to
to influence
products
and uh... we're not we can't go into the details of what that would mean for
manufacturing steel a two separate class
but basically there's got to be away
to a chance to meet you hardness by adding various different kinds of
elements different
uh... different kinds of metals
to the steel
and so then any natural question would be if you've got an order that
everything you make or at least a lot of what you make has to exceed a hardness
of sixty eight
how do you have to choose the means of that you've got a ninety percent
probability that randomly chosen brought
half-hearted soffit east sixty eighth well
that would mean
that we
interest in the probability that the hardness is greater than sixty eight
but we have to set table two point ne
well still same story this attractive you define by the signup
it's just that the so we don't know what the new windows right
and well we don't know what the new with you just keep it as a variable
and
well we have an equation here that now says that eight-state ability
distributed at their home
needs to the probability that that thing is greater than the summer is supposed
to be point nine
well let's take a look at the table
is
the entry point nine
is sketchy
somewhere around here right
that's about point nine students point eight nine nine seven which is the chief
or
one point two and ten one point to eight apparently
and so you know that that's what we
have to find here and okay i think made a mistake with the inequality
we give the area to the right afar number is two point nine rail
that would mean that the area
to the left of the negative number is supposed to be point nine
and now we get the one point two agents of the problem that you don't see being
smaller than one point
to weighs approximately point eight nine nine said we can get an exact match at
least with a table in fact
typically overkill anyway
but basically what that means is that megan sixty-eight minus mu divided by
two
needs to be
equal to
approximately
one point two eighteen personal any question that the consult
so
made sixty eight months nita bug but you
we set that equal to one point two eight
that means sixty eight months newest
uh... made it's two point five six
and then we
to crack
uh... we have u_n_ at the two point five six million up with new being
sixty-eight plus two point five six well thats
seventy point five six and as a quick sent the check
if we want it the area to the right off sixty eight is about point nine and we
definitely need
amy and that is to the right
of sixty eight in seventy
point five six
fairly quickly
all key so
uh... something that you made for the homework if you've got a standalone
abandoned variable
then for any also that means zero and when the numbers the alpha denotes the
value such that the probing to see greater than video clip is
alpha
that is althought the area under the standard normal curve lies to the right
of c l_ friends of the picture is
if this year here's elsa
then this cut off
z l two will meet these values when we talk about
uh... confidence intervals and it's basically one of those things for a lot
of things and
especially also for the accustomed to working with
punitive distributions were interested in the area to the left of the number
well
here we're talking now about the area
to the right of the number which every so often
we also need
okay final properties
if you could only distributed randi rhodes paramus new u_n_ sigma
then the following properties hold the probability to that excuse
uh... within one standard deviation of the mean
is point six eight the productivity exits with into a standard deviations of
the knee
it's about point nine five in the program that exits
within three standard deviations of means about
point nine nine seconds you may have
scene
these probabilities sixty eight percent ninety five percent
ninety-nine point seven percent you may have
seen or heard about these probabilities
in matches before because basically when you know that you compute your
uh... do-it-yourself elaboration into your standard air
people often say quote not quite correctly that you've got a sixty eight
percent chance that your actual value is
uh... within one standard eric would you measured ninety five percent chance it's
within two standard
errors and i may nine point seven percent chance that its
within three senator surrounds is something we're starting to connect
two steps that you may well have to
already seen in last place indeed this is the
probabilistic justification
into order
fully understand what the plan which means we will talk about
confidence intervals
but basically
that's just talk about rabies
ka of where these
profanities come from that come from the fact that there is a certain reasonable
assumption that that certain things that you measuring the map
uh... normally distributed
in uh... well then
that means that we would really end up with these kinds of probabilities
into the way you can get these problems the sake thirty simply interest in the
probability that new minus k statements was index is smaller than you could ask
a signal which is exactly what we have you with people's one two and three
well then use attract new divide by sigma right and that means here
excellence new yorkers not
is a z
and from the other computations here only keeping
a minus k on the left
indicate on the right
so that that's just straight out of her
and i'll i'll that stand with the transmitting at this probity of see
small i think a month's priority of the small negative k which is a broad if
he's like a minus
one minus the probably didn't see smaller than came right there
where just using the threatening than that
complimentary probability
in that still problems using my zeese one of them came on this one and if we
know just work out the numbers
which we can look up in the table
we end up with approximately point six eight for people's one we end up with
approximately point nine five four p stoop and we end up with approximately
point nine nine seven
forty three
and uh...
that basically prove
that these probabilities freely are
what they are
claimed to be
uh... in so that means in this is really just a picture that we're finishing with
uh... in eighty normal distribution to have a sixty eight percent chance to be
within one standard deviation of the main
eight ninety five percent chance to get an outcome that is
within two standard deviations of the me
iti
ninety nine point seven percent chance to
getting outcome tickets with him
three standard deviations off the main
we will
uh...
we will
the come a lot more unfamiliar with
normal distributions ensuite
continue because
uh... they will come back to us for example in distant central limit theorem
uh... the numbers themselves are right now just background information date
are apparent date that they have something to do
we've things that you measure in the lab we'll see what the exact connections
later on
for now
you have uh... three
uh... three families all ralph probability density functions that you
or i think you can compute probabilities four
in cell
let's get to that extent you compute simple cookies